GALOIS COHOMOLOGY ANDNUMBER FIELD COUNTING

By

Brandon Alberts

A dissertation submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

(Mathematics)

at the

UNIVERSITY OF WISCONSIN – MADISON

2018

Date of final oral examination: August 7, 2018

The dissertation is approved by the following members of the Final Oral Committee:

Professor N. Boston, Professor, Mathematics

Professor M. Matchett Wood, Professor, Mathematics

Professor J. Ellenberg, Professor, Mathematics

Professor S. Marshall, Assistant Professor, Mathematics

i

Abstract

Arithmetic Statistics is the study of how arithmetic objects are distributed. One of the

most popular examples in this subject are number field counting problems. Initially dis-

cussed by Cohen-Lenstra and Malle, there are now a variety of conjectural heuristics and

solutions describing the distribution of number fields with prescribed Galois group, local

behavior, degree, etc when ordered by their discriminant. Despite the wealth of com-

putational and theoretical evidence, very few cases have been proven. This dissertation

details the author’s work on proving new cases of a few number field counting problems.

Namely, we prove some new cases of Wood’s nonabelian Cohen-Lenstra heuristics and set

the stage to prove more, and we prove new upper bounds for the weak form of Malle’s

conjecture which agree with Malle’s predicted bounds conditional on the conjectural

growth of `-torsion in class groups of number fields with fixed degree.

ii

Acknowledgements

I have been a graduate student at the University of Wisconsin-Madison for five years,

and I never would have made it this far without every single one of the people in my

life. Thank you all. I would never be able to list everyone, but I do want to extend a

special thanks to the following people:

My advisor Dr. Nigel Boston, who essentially taught me how to do research math-

ematics for the past five years, as well as the other members of my defense committee

Dr. Melanie Matchett Wood, Dr. Jordan Ellenberg, and Dr. Simon Marshall. Your

willingness and eagerness to interact with graduate students like myself has had a huge

impact on me and has been immensely helpful in completing my graduate studies.

My peers in the graduate program at UW-Madison. Anyone who could see us would

never doubt that mathematicians are one of the most social groups on the planet. Your

aggressive friendship made moving to Madison easy and leaving hard.

Dr. Jeanne Wald and the Advanced Track program at Michigan State University.

You showed me that mathematics was worth pursuing as a career and rescued me from

a life as an engineer. Everyone who was a part of that program made it possible for me

to go to graduate school in the first place with the best undergraduate education I could

ever want.

The SuperfriendsTM , as they like to call themselves for some reason. You are the

first group of best friends I’ve ever made after family, and despite the fact that I’ve been

away at graduate school nearly twice as long as we attended Michigan State together I

still feel at home when I come back to visit.

iii

My family, without whom I never would have fallen in love with mathematics in the

first place. Thank you for being my first teachers, and for supporting me and visiting

me as my studies have taken me farther from home. You mean the world to me.

My coauthor, Dr. Jack Klys. Together we took our first steps into collaborative

mathematics, teaching me things about research I could’ve never learned in school by

myself.

Everyone who has given mathematical advice or provided feedback on the work which

appears in this dissertation, especially Dr. Nigel Boston, Dr. Melanie Matchett Wood,

Dr. Jordan Ellenberg, Dr. Simon Marshall, Dr. Jurgen Kluners, Dr. Jacob Tsimerman,

Dr. Jack Klys, Yuan Liu, Soumya Sankar, and anonymous referees.

The National Science Foundation, which provided me additional financial support

via grant DMS-1502553 to the UW-Madison Mathematics Department.

iv

Contents

Abstract i

Acknowledgements ii

1 Introduction 1

1.1 How do we count number fields? . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Cohen-Lenstra Heuristics . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Malle’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Embedding Problems . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Fouvry-Kluners Sieve . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Nonabelian Cohen-Lenstra Moments 25

2.1 Admissible Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.1 GI-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.2 A family of groups without GI-extensions . . . . . . . . . . . . . . 30

2.2 Quaternion Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.1 The Expected Number . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 The Asymptotic Count . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.3 A Corresponding Distribution . . . . . . . . . . . . . . . . . . . . 63

2.3 Other Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3.1 Dihedral Group of Order 8 . . . . . . . . . . . . . . . . . . . . . . 66

v

2.3.2 Centrally Admissible Pairs pG,G1q . . . . . . . . . . . . . . . . . . 68

2.3.3 Example: Heisenberg Groups . . . . . . . . . . . . . . . . . . . . 82

2.3.4 Admissible Pairs of the Form pG,G C2q . . . . . . . . . . . . . 87

3 Number Field Counting 96

3.1 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1.1 Group Theoretic Lemmas . . . . . . . . . . . . . . . . . . . . . . 98

3.1.2 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.1.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A The Quaternion Group 117

A.1 The Asymptotic Count: Remaining Imaginary Cases . . . . . . . . . . . 117

A.1.1 The case d 0, d 4 mod 8 . . . . . . . . . . . . . . . . . . . . 117

A.1.2 The case d 0, d 0 mod 8 . . . . . . . . . . . . . . . . . . . . 124

A.2 The Asymptotic Count: Real Cases . . . . . . . . . . . . . . . . . . . . . 131

A.2.1 The case d ¡ 0, d 1 mod 4 . . . . . . . . . . . . . . . . . . . . 132

A.2.2 The case d ¡ 0, d 4 mod 8 . . . . . . . . . . . . . . . . . . . . 137

A.2.3 The case d ¡ 0, d 0 mod 8 . . . . . . . . . . . . . . . . . . . . 142

Bibliography 148

1

Chapter 1

Introduction

As the introduction to my dissertation, it would be an immense pleasure to walk the

reader through the process by which I entered into research mathematics. I say it would

be an immense pleasure to do, but I fear it would be an immense pain to read. My

graduate studies were not linear, like a good textbook or survey paper. They were

jagged, each time I learned something new I would revisit it at a later time to find my

understanding inadequate. If I were to condense that process into one paper, I suspect

I would confuse the reader by repeatedly backtracking. Such a paper would be littered

with retractions and modifications of statements, analogies, and intuitions merely a few

pages old, something only a writer more skilled than myself could get away with.

Instead, I want to present this introduction as a snapshot of my current understand-

ing of the subject. It is my hope that not only will this provide the best background

leading up to the presentation of my original work conducted as a graduate student, but

also that it will serve as a kind of “end product” of my graduate education and make

this dissertation a fitting conclusion to my formal studies. At the same time, the end

of my formal studies marks just one stage of a lifetime of study of mathematics. With

any luck, in a few years time I will revisit this dissertation and find that I have again

relearned the material and ideas presented here in a deeper way.

2

1.1 How do we count number fields?

At some point in my graduate career, I was describing part of my research to Jordan

Ellenberg. One of my very first original results was proving that the expected number

of unramified H8-extensions of imaginary/real quadratic fields is infinite. Jordan asked

me if this was the “same kind of infinity” as one gets from genus theory, when finding

the expected number of unramified C2-extensions of imaginary/real quadratic fields.

I find that this question really hits at the heart of number field counting, and problems

in arithmetic statistics in general, where just knowing that something is infinite does not

satisfy us. We want to know what kind of infinity, a question which we often answer by

showing the objects being counted fit certain distributions or have a certain asymptotic

growth.

Specifically in response to Jordan, my answer was “no”. In joint work with Jack

Klys, we improved my original result by proving that the number of unramified H8-

extensions of imaginary/real quadratic fields with absolute discrimintant bounded by X

is asymptotic to cXplogXq2 for some explicit constant c as X tends to 8. This is

a strictly faster growth rate than the one arising from genus theory, which shows that

the number of unramified C2-extensions of imaginary/real quadratic fields with absolute

discriminant bounded by X is asymptotic to c1X logX for some explicit constant c1 as

X tends to 8.

Given a family of number fields F , we can talk about how big this family is by

measuring the asymptotic growth of #tK P F : DKQ Xu, where DKQ is the absolute

discriminant. Each of these values is finite, as there are only finitely many number fields

whose discriminant is bounded, and when we let X tend to infinity they grow at some

3

rate which we can sometimes describe by an elementary function. For example, consider

Q the family of real quadratic fields. Quadratic fields are in one-to-one correspondence

with quadratic discriminants, i.e. d n, 4n, or 8n where n P Z is odd, squarefree, and

such that n 1 mod 4, n 3 mod 4, and n 1 or 3 mod 4 respectively. Then we

can count real quadratic fields by the following method:

#tK P Q : discpKq Xu #t0 n X : n 1 mod 4 odd squarefreeu

#t0 4n X : n 3 mod 4 odd squarefreeu

#t0 8n X : n odd squarefreeu

1

2#t0 n X : n odd squarefreeu

1

2#t0 4n X : odd squarefreeu

#t0 8n X : n odd squarefreeu

1

2

1 1

2

16

π2X 1

2

1 1

2

16

π2

X

4

1 1

2

16

π2

X

8

3

π2X

There are infinitely many real quadratic fields, but this statement in a certain way

answers “what kind of infinity”, by giving us information about how these fields are

distributed when they are ordered by discriminant (in this case, roughly linearly).

This kind of counting is central to arithmetic statistics, and the two conjectures that

we will discuss: Cohen-Lenstra heuristics and Malle’s conjecture.

4

1.1.1 Cohen-Lenstra Heuristics

Given a number field K and an odd prime p, let ClpKq denote the class group of K and

ClppKq denote the p-part of the class group. Cohen and Lenstra [13] defined distribu-

tions on finite abelian p-groups given by µCLpAq proportional to 1|AutpAq| and µCLpAqproportional to 1p|A||AutpAq|q. They provided experimental and theoretical evidence

that the p-parts of the class groups of imaginary and real quadratic fields are distributed

according to µCL. More explicitly, if we let Q be the family of imaginary()/real()

quadratic fields respecitively, then Cohen-Lenstra [13] conjecture that

limXÑ8

#tk P Q : Clppkq A,DkQ Xu#tk P Q : DkQ Xu µCLpAq.

This statement as a distribution is slightly different than a number field counting state-

ment, but we can rephrase it using class field theory. The class group is isomorphic to

the Galois group of the maximal unramified extension, and so an equivalent formulation

for Cohen-Lenstra’s heuristics (often called the moments version) is that for every finite

odd abelian group A

limXÑ8

#tKk : k P Q, Kk unramified,GalpKkq A,DkQ Xu#tk P Q : DkQ Xu 1

|AutpAq|limXÑ8

#tKk : k P Q, Kk unramified,GalpKkq A,DkQ Xu#tk P Q : DkQ Xu 1

|A||AutpAq|

Inspired by this expression, we can make the following definition as in Wood [59] or

Bhargava [6]:

Definition 1.1. Let Gurk denote the Galois group of the maximal unramified extension

5

of a number field k. For a finite group G, define

EpGq limXÑ8

¸kPQ,DkQ X

|SurjpGurk , Gq|¸

kPQ,DkQ X1

where Surj refers to the set of surjection homomorphisms.

Remark: This is the same as counting number fields, up to a factor of |AutpGq|.

Conjecture (The Moments Version of Cohen-Lenstra heuristics). For any finite odd

abelian group A, EpAq 1 and EpAq |A|1.

In this form, Cohen-Lenstra heuristics have only been verified for a single finite odd

abelian group, namely A C3 as a consequence of Davenport-Heilbronn [15]. They are

widely believed, in part because a similar statement is shown to hold over function fields

by Ellenberg-Venkatesh-Westerland [22].

There have been various works generalizing Cohen-Lenstra heuristics to predict or

provably compute the number of unramified G-extensions of quadratic fields when G is

an arbitrary group. The first case one might try to tackle is for finite abelian 2-groups,

but genus theory implies that EpC2q 8. Gerth [27] proposed the use of 2Cl2pkqinstead to avoid genus theory, in which case the predicted values of EpAq agree with

the predicted values for odd groups. This was proven for C2 (relating to the 4-rank of

the class group) by Fouvry-Kluners [25], and results on the 2k-rank of the class group

consistent with this conjecture were proven in a recent paper by Smith [54].

Boston-Bush-Hajir [11] generalize Cohen-Lenstra heuristics to predict the distribu-

tion of unramified G-extensions of quadratic fields when G is a p-group for an odd prime

p. This predictions agree with Cohen-Lenstra’s original predictions after abelianizing.

6

Wood [59] generalizes Cohen-Lenstra moments to arbitrary finite groups. She divides

up her counting functions a little more via pairs of groups pG,G1q. She counts unramified

extensions Lk such that GalpLkq G and GalpLQq G1, which we call an unramified

pG,G1q-extension for simplicity. The following definition was used by both Wood [59]

and Bhargava [6]:

Definition 1.2. Given a pair of finite groups pG,G1q define

NpG,G1;Xq #tLk : k P Q, Lk unramified pG,G1q-extension, DkQ Xu

and

EpG,G1q limXÑ8

NpG,G1;Xq#tk P Q : DkQ Xu

An “admissible pair” of groups is defined by Wood [59] to be a pair of groups pG,G1qfor which there could exist an unramified pG,G1q-extension of a quadratic field. Group

theoretically, this means G1 ¤ G o C2 such that if N G G ¤ G o C2 and C2 xσythen pp1, 1q, σq P G1, G1 XN ¤ G G surjects onto the first coordinate, and G1 XN is

generated by elements inverted by conjugation with pp1, 1q, σq. For example, the only

admissible pair pG,G1q for which G is abelian is given by G1 GC2 where σpgq g1

for any g P G. This definition is made in such a way that NpG,G1;Xq 0 for any

nonadmissible pair pG,G1q. For admissible pairs, Wood makes the following conjecture:

Conjecture (Wood’s Conjecture [59]). For any admissible pair pG,G1q let c be the

number of conjugacy classes of elements g P G1G of order 2. If pG,G1q is “good”, i.e.

c 1, then

EpG,G1q |H2pG1, cqr2s||AutG1pGq| and EpG,G1q |H2pG1, cqr2s|

|c||AutG1pGq|If pG,G1q is “not good”, i.e. c ¡ 1, then EpG,G1q 8.

7

H2pG1, cq is a quotient of the Schur multiplier H2pG1,Zq and AutG1pGq is the subgroup

of AutpGq fixing G1 ¤ G o C2 setwise.

Wood’s conjecture, also called the nonabelian Cohen-Lenstra moments, has been ver-

ified in more cases that the original moments version of the Cohen-Lenstra heuristics. In

Wood’s original paper [59] she proves an analogous statement over function fields sim-

ilar to the work of Ellenberg-Venkatesh-Westerland [22] on the original Cohen-Lenstra

heuristics. Prior to Wood’s conjecture, Bhargava [6] proved the first results of this kind

for pAn, Snq and pSn, Sn C2q when n 3, 4, 5.

In Chapter 2, we discuss the author’s work on this conjecture and related problems.

This work is also presented in stand-alone papers [1] [2] and in joint work with Jack Klys

[4]. This chapter begins with a discussion of admissible pairs pG,G1q with rG1 : Gs 2

in Section 2.1, where we classify when there exists such a pair for G a group of affine

transformations over a finite field (see Theorem 2.12). Section 2.2 addresses Wood’s

conjecture directly for the unique admissible pair pH8, H8 C2q, where H8 refers to the

quaternion group of order 8. We prove in Theorem 2.14 that EpH8, H8 C2q 8 as

predicted by Wood. Then, in joint work with Klys, we prove Theorem 2.23 which gives

an asymptotic count:

NpHk8 , H

k8 C2;Xq 1

|AutHk8C2

pHk8 q|4k

¸dPD

X

3kωpdq

OX plogXq3k2ε

NpHk8 , H

k8 C2;Xq 1

|AutHk8C2

pHk8 q|24k

¸dPD

X

3kωpdq

OX plogXq3k2ε

Remark: A Tauberian theorem can be used to show°

3kωpdq XplogXq3k1. This

shows that the error term is really an error term, but moreover this kind of asymptotic

bridges the gap between Cohen-Lenstra heuristics and Malle’s conjecture (see Section

8

1.1.2). Wood gives some evidence that in the function field case we should expect this

asymptotic to be XplogXqc1, which matches this result. Klys [39] generalizes this

result to a subset of groups G which are a central extension of Cn2 by C2.

We then conclude the chapter with results in Section 2.3 that begin a program of

extending these results to other families of groups. This includes admissible pairs pG,G1qwhere G1 has nilpotency class 2 in Section 2.3.2 and where G1 GC2 in Section 2.3.4.

1.1.2 Malle’s Conjecture

Let K be a number field. Instead of counting unramified extensions of a family of num-

ber fields like in Cohen-Lenstra heuristics, Malle specifically studied counting degree

n extensions LK inside of a fixed algebraic closure K when ordered by absolute dis-

criminant in [42] and [43] with no restrictions on local conditions, or equivalently by

the absolute norm of the relative discriminant. The Galois group of LK (or rather,

of the Galois closure of LK) is GalpLKq Sn a transitive subgroup acting on the n

embeddings L ãÑ K. Let G Sn be a transitive subgroup and

NpK,G;Xq #tLK|GalpLKq G, |NKQDLK | Xu

Malle makes the following conjecture:

Conjecture (The Strong Form of Malle’s Conjecture). Let G ¤ Sn be a transitive

subgroup. Then

NpK,G;Xq cpK,GqX1apGqplogXqbpK,Gq1

where apGq mingPG indpgq, indpgq n #torbits of g P Snu, and bpK,Gq |tg PG|indpgq apGqu | where the relation is given by conjugation by G and the action by

the cyclotomic character χ : GalpKKq Ñ GalpQabQq pZ given by δ.g gχpδq.

9

This is often referred to as the strong form of Malle’s conjecture. It has been verified

for G an abelian group in any representation by Wright [60], G Sn for n 3 by

Datskovsky-Wright [14] and n 4, 5 by Bhargava-Shankar-Wang [8], S3 S6 in its

regular representation over Q by Bhargava-Wood [9], Sn A Sn|A| for A S|A| an

abelian group in its regular representation with p|A|, n!q 1 for n 3, 4, 5 by Wang [56],

and C2 o H for certain groups H by Kluners [36]. Similar results are known when the

extensions LK are ordered by other invariants, for example Wood proves the analogous

result for abelian groups ordered by conductor [58].

Unfortunately, the conjecture is not true in this form, as Kluners [34] provided a

counter-example for G C3 o C2 S6 for which bpK,Gq is too small. There have been

proposed corrections for bpK,Gq by Turkelli [55], but the 1apGq exponent is still widely

believed to be correct. This leads to the weak form of Malle’s conjecture:

Conjecture (The Weak Form of Malle’s Conjecture). Let G ¤ Sn be a transitive sub-

group. Then

X1apGq ! NpK,G;Xq ! X1apGqε

where we say fpXq " gpXq if lim supXÑ8 fpXqgpXq ¡ 0. This has been proven

in more cases, notably by Kluners-Malle [37] for G any nilpotent group in its regular

representation G S|G|. Kluners-Malle also proved the upper bound for `-groups in

any representation, and all of their results remain true if we restrict to fields unramified

away from a finite set S of places. Additionally, for Dp the dihedral group of order 2p

for a prime p Kluners proved the lower bound unconditionally and the upper bound

conditional on Cohen-Lenstra heuristics [35]. There are no known counterexamples to

the weak form of Malle’s conjecture.

10

Aside from the cases in which the weak form of Malle’s conjecture is not known, there

has been work on proving upper bounds that are not quite as sharp as X1apGqε. The

subject of studying just this upper bound is a vibrant area itself, and centers on a folklore

conjecture attributed to Linnik: Let NK,npXq be the number of extensions LK of degree

n with the absolute norm of the relative discriminant bounded |NKQpdiscpLKqq| X.

Then

NK,npXq CK,nX

Linnik’s conjecture would follow from the strong form of Malle’s conjecture, while the

weak form of Malle’s conjecture would imply logNK,npXq logX. Progress towards

this conjecture has been slow; the best general bounds are due to Schmidt [52], which

state

NK,npXq ! Xn2

4

This was improved upon by Ellenberg-Venkatesh [23] for large n, who proved that there

exist constants An depending on n and an absolute constant C such that

NK,npXq !XDn

KQArK:Qsn

exppC?lognq

which shows in particular that

lim supXÑ8

logNK,npXqlogX

!n nε

Keeping in step with the philosophy of Malle’s conjecture, Dummit [19] proved an upper

bound which improves upon Schmidt’s bounds when G Sn is a proper transitive

subgroup. If any subgroup G1 ¤ G containing a point stabilizer has index at most t P G,

then Dummit shows

NpK,G;Xq ! X1

2pntqp°n1i1 degpfi1q 1

rK:Qsqε

11

for f1, ..., fn a set of primary invariants of G for which deg fi ¤ i. This result significantly

improves upon the bounds given by Schmidt in many cases, but is still not very close

to the bound predicted by Malle. In particular, this exponent is larger than 12 1

2nrK:Qs

which is very close to 12 for large K, although many groups have 1apGq 12 (namely

all groups in the regular representation other than G C2).

When one asks the analogous question over function fields FqpT q, a preprint of

Ellenberg-Tran-Westerland [21] shows that when q " |G| Malle’s predicted upper bound

is satisfied. This gives strong evidence that Malle’s predicted upper bounds should hold

over number fields, although the methods used for function fields do not appear to

transfer to number fields.

In Chapter 3 we discuss the author’s work on new upper bounds for NpK,G;Xqwhen G is a solvable group. This work is also presented in a stand-alone paper [3]. We

prove conditional upper bounds depending on `-torsion in class groups:

Theorem 1.3. Fix a normal series

t1u G0 ¤ G1 ¤ ¤ Gm1 ¤ Gm G

with Gi G such that the factors GiGi1 are nilpotent. Then lim supXÑ8 logpNpK,G;Xqq logX

is bounded above by

1

apGq

1

m1

i1

NipEi 1qEi

¸` prime

ν` p|GiGi1|q lim suprL:Ks¤Ni,DLQÑ8

logp|ClpLqr`s|qlogpDLQq

where Ni |pGGi1qCGpGiGi1q|, Ei is the order of the largest cyclic group in

pGGi1qCGpGiGi1q, and ν`pnq is the power of ` in the prime factorization of n.

We can make a few immediate observations about this theorem.

12

1. If G is a nilpotent group, then the trivial subnormal series t1u G can be used

in this theorem with m 1. When m 1, the summation is trivial and we

immediately prove the upper bound for the weak form of Malle’s conjecture:

lim supXÑ8

logpNpK,G;XqqlogX

¤ 1

apGq

2. Minkowski’s bounds imply a trivial bound for the size of the class group among

fields L of bounded degree rL : Ks, namely |ClpLqr`s| ! D12εLQ . This implies an

unconditional upper bound

lim supXÑ8

logpNpK,G;XqqlogX

¤ 1

apGq

1

m1

i1

NipEi 1q2Ei

Ω p|GiGi1|q

where Ωpnq °p νppnq. Typically this is not a very good bound, and will be

much larger than Schmidt’s or Dummit’s bounds. However, in select cases when

apGq " m we get significant improvements. For example, when G Dp Sp is

the dihedral group for p an odd prime, then apGq 2p1

and m 2. Plugging in

all the relevant information gives the bounds:

lim supXÑ8

logpNpK,Dn;XqqlogX

¤ 3

p 1 3

2apGqThese are significantly smaller that Schmidt’s bound of p2

4, and Dummit’s bound

which is at least 12 1

2prK:Qs 12. In fact, this bound realizes the unconditional

bounds proven for Dp by Kluners [35]. In many ways, Theorem 1.3 can be veiwed

as a generalization of Kluners’ results to other solvable groups. See Section 3.1.3 for

data comparing the bounds obtained by this theorem to Dummit’s and Schmidt’s

bounds for small solvable G.

3. These bounds improve with any power saving on the bound for `-torsion of class

groups with fixed degree. A recent result of Ellenberg-Venkatesh [24] gives the

13

largest known power savings, namely that for fields rL : Ks d GRH implies

|ClpLqr`s| ! D1212`pd1qεLQ

Conjecturally the `-torsion of the class groups of degree d number fields is bounded

by DεLQ (as discussed in Brumer-Silverman [12], Duke [17], Ellenberg-Pierce-Wood

[20], and Zhang [61]). For the bounds in our theorem, every term in the summation

which satisfies this conjecture is zero. For any transitive solvable group G Sn, if

|ClpLqr`s| ! DεLQ for fields rL : Ks ¤ Ni, then we achieve the upper bound from

the weak form of Malle’s conjecture:

lim supXÑ8

logpNpK,G;XqqlogX

¤ 1

apGq

We will in fact prove a more general result than Theorem 1.3. We will prove anal-

ogous bounds for number fields with restricted local behavior at any number of places.

Such results appear to be absent from much of the literature cited in this introduction,

although in many cases the proofs are not significantly different. For example Kluners-

Malle [37], Wright [60], and Wood [58] all discuss cases with certain restricted local

behaviors at finitely many places which do not behave significantly differently (with the

exception of those cases that fall under the Grunwald-Wang Theorem).

Additionally, we will prove the analogous result to Theorem 1.3 under different order-

ings of the number fields. While we have classically been interested in counting number

fields when ordered by the discriminant of a subfield, more recently there has been inter-

est in studying more general invariants such as the conductor or the product of ramified

primes. In certain cases, these invariants have nicer properties than the discriminant

and can be easier to work with. Wood [58] counts abelian extensions ordered by con-

ductor, and shows some ways in which this invariant is nicer than the discriminant.

14

Bartel-Lenstra [5], Dummit [18], and Johnson [33] continue this philosophy by studying

different questions when ordering number fields by various invariants. We will cater to

this perspective, and prove the analog to Theorem 1.3 when ordering by a wide family

of invariants.

1.2 Background Material

Progress has been made on number field counting problems via a variety of differ-

ent methods. When studying the function field analogs of such problems Ellenberg-

Venkatesh-Westerland [22] and Wood [59] take advantage of the connection to geometry

of curves. Kluners and Malle in various papers [35] [36] [37] adapt methods from solu-

tions to the inverse Galois problem. Bhargava [6] recently introduced a new approach

generalizing Gauss’s composition laws to other n-ary k-ic forms by counting lattice points

on moduli spaces.

The common thread of this paper is cohomological methods, which most closely

align with those methods used by Kluners and Malle and have proven very amenable

to standard analytic techniques. We will briefly discuss some of the cohomological and

analytic results that have been used for these problems in the past that we will reference

in the main body of the paper.

1.2.1 Embedding Problems

Let GQ GalpQQq denote the absolute Galois group of Q with Dp and Ip the cor-

responding decomposition and inertia groups at the prime p (defined as subgroups of

GQ up to conjugacy for p finite or infinite). Given an extension E of a group G by an

15

abelian group A, suppose we have a commutative diagram as follows:

GQ

1 A E G 1

frfπ

Then the embedding problem asks when f lifts to a continuous homomorphism rf : GQ ÑE that makes the diagram commute. This problem is “solved” in a certain sense in the

case of A abelian and E a central extension, in that a solution exists if and only if a

solution exists to the corresponding local embedding problems:

Dp

1 A E G 1

fprfpπ

where fp f |Dp [44] [53]. The proof is a direct consequence the following result:

Lemma (Lemma 2.1.5 in Serre’s Topics in Galois Theory [53]). For ` a prime and GQ

acting trivially on C`, the restriction map

H2pQ, C`q ѹp

H2pQp, C`q

is injective.

The solution to the embedding problem for central extensions gives possibly the

slickest solution to the inverse Galois problem for nilpotent groups. All nilpotent groups

have a composition series which is central (i.e. a composition series G0 ¤ Gm such

that Gi Gm and GiGi1 belongs to the center of Gi1Gi1), so Kronecker-Weber

combined with the solution to the embedding problem for central extensions allows one

to construct a Galois extension with any finite nilpotent group as its Galois group. See

Serre [53] for a more detailed discussion. The embedding problem plays a role in the

16

work of Kluners-Malle [37] in proving the weak form of Malle’s conjecture for nilpotent

groups in the regular representation.

In Section 2.3.2, it will be of interest to ask about solutions to the embedding problem

with prescribed local behavior. We call a lift rf unramified at p over f if ker fXIp ¤ ker rf(and unramified at all places if this is true fro all places p (finite or infinite)). Let

K Qker f, L Qker rf

, and Kp, Lp the completions at a prime above p. Then rffis unramified if and only if LK is unramified, i.e. if and only if rf factors through

GalpKur,abKq where Kur,ab is the maximal unramified abelian extension of K, otherwise

known as its Hilbert class field.

The solutions to the embedding problem for central extensions are a priori indepen-

dent of ramification. With a little extra work, we can determine when a solution can be

found with specific ramification. Some results of this nature in more specific instances

were proven in [45] [46] [47] by Nomura and some general results can be found in [53].

We will present the form of these results that we will use in this dissertation, as well as

their proof:

Theorem 1.4. Given f : GQ Ñ G a continuous homomorphism with fpDpq abelian,

there exists a solution to the embedding problem rf : GQ Ñ E unramified at finite places

if and only if ResGfpDpqprEsq is abelian and ResGfpIpqprEsq 0 for all places (finite and

infinite). Here, Res is the restriction map from group cohomology and rEs is the class

of the extension E in H2pG,Aq.

Proof. For the forward direction, we have that fpIpq Ippker fXIpq Ippker rfXIpq rfpIpq. This implies that there is a section τp : Ip Ñ π1pIpq, forcing ResGfpIpqprEsq 0.

Additionally, we have LpKp is unramified, with rfpDpq GalpLpQq and fpDpq

17

GalpKpQq abelian. Any unramified extension of an local field abelian over Qp is also

abelian over Qp by a standard result in local class field theory, so rfpDpq is abelian. Then

ResGfpDpqprEsq has an abelian subgroup which surjects onto fpDpq. Since this is a central

extension of fpDpq this implies that ResGfpDpqprEsq must also be abelian.

For the converse, note that ResGfpIpqprEsq 0 implies the local embedding prob-

lem is solvable at all places, so there exists a possibly ramified lift rf with π rf f .

ResGfpIpqprEsq 0 implies that rfpIpq is a trivial extension of fpIpq. ResGfpDpqprEsq abelian

implies that for p 2 inertia factors through procyclic Zp with finite image for both f

and rf . Therefore we must have rfpIpq Cnp fpIpq with pnp, |fpIpq|q 1. For p 2

we get a similar result with possibly an extra factor of C2. Let ρp : Ip Ñ A be rf |Ipcomposed with projection onto the first coordinate Cnp (or possibly C2Cn2 in the case

p 2) and embedded into A. Define the map

ρ pρpq : GQ ¹

Iabp Ñ A

This produces a map

g f ρ : GQ Ñ G A

which satisfies gpIpq rfpIpq since p|fpIpq|, |ρpIpq|q 1 by |ρpIpq| np or 2n2. Then the

map rf ρ : GQ Ñ E A composed with π gives g, and composed with projection onto

the first coordinate gives rf . Then

pA Aq X p rf ρqpIpq pρp ρpqpIpq

is a diagonal embedding of Ip into AA ¤ EA. This implies that pAAqXp rfρqpIpq ¤∆ the diagonal subgroup. Let π∆ : E A Ñ E be the quotient map by ∆ (in other

18

words, we are adding the extensions rEs and rGAs together in H2pG,Aq, which gives

back rEs). It then follows that π∆ p rf ρq : GQ Ñ E is a morphism with

ππ∆p rfpxq, ρpxqq πp rfpxqρpxq1q

fpxq

and

AX π∆p rf ρqpIpq 0

which implies that if ker f X Ip ¤ kerπ∆p rf ρq so that it is unramified at all finite

places.

We can also ask that rf be unramified at the infinite place. This is trivially true if f

is ramified at 8 or if 2 - |A|. Moreover, if 2 | fpIpq for some finite prime p 3 mod 4,

then we can take g : GQ Ñ GalpQp?pqQq and combine it with rf ramified at infinity in

the same way as in the proof of this theorem to get a new lift unramified at infinity. As

for when 2 | |fpIpq| only for primes p 1 mod 4, the question becomes more difficult.

Lemmermeyer addressed this when he studied unramified H8-extensions of quadratic

fields [40], but when studying the expected number of such extensions the author in

joint work with Klys avoids using this result by showing that it only contributes to the

error term [4] (see the Appendix for the proof). Although we do not address ramification

at 8 in this section, we expect the methods used by Lemmermeyer to extend to more

general cases without much difficulty.

Define Frobp to be a lift of the Frobenius automorphism in GalpFpFpq to GQ, defined

up to conjugacy and modulo Ip. Then we have the following corollary:

19

Corollary 1.5. Given f : GQ Ñ G with fpDpq abelian, and rEs P H2pG,Aq such that

ResGfpIpqprEsq 0, then there exists a lift rf : GQ Ñ E unramified at finite places if and

only if rfpFrobpq, fpIpqsE 0 where r, sE : GGÑ E is the commutator of E.

Proof. E is a central extension of G, so the commutator factors through GG. Notice

that rag, bgsE ra, bsgE respects conjugation, so WLOG we can fix a representative of Dp

and Ip up to conjugation. Suppose x, y P Dp are both representatives of Frobenius Frobp.

Because fpDpq ¤ G is abelian, it follows rfpxq, fpIpqsE, rfpyq, fpIpqsE kerπ A.

Suppose rfpxq, fpIpqsE 0. Then rfpIpq, fpIpqsE 0 since ResGfpIpqprEsq 0 implies

π1pIpq is a trivial extension of the abelian fpIpq. Then for any i P rfpIpqr rfpxq, isi rfpxq rfpxqi

rfpyq rfpy1xqi

rfpyqi rfpy1, xq

r rfpyq, isi rfpyq rfpy1xq

r rfpyq, isi rfpxqnoting that rfpy1xq P rfpIpq. Thus rfpFrobpq, fpIpqsE is a well-defined subgroup of A.

ResGfpDpqprEsq is abelian if and only if a representative of Frobenius commutes with

inertia (i.e. rfpDpq abelian), and so the result follows from the previous theorem.

1.2.2 Fouvry-Kluners Sieve

At the time of this dissertation, there have been several papers utilizing the Fouvry-

Kluners sieve in different scenarios. This includes, but is not limited to, A.-Klys [4] (see

Section 2.2.2), Klys [38], and Fouvry-Kluners [25] on the average values of the 4-rank

20

of the class group of quadratic fields. Each of these papers contains a summary of how

the sieve proven by Fouvry-Kluners [25] applies to other scenarios. Using some graph

theory, we can reframe Fouvry’s and Kluners’ sieve and give a concise general statement

which is readily applied to new problems.

For any finite set of vertices V and a function Φ : V V Ñ F2, define GrpΦq to

be the (undirected) graph with vertex set V and edge set EpΦq tpu,vq P V V :

Φpu,vqΦpv,uq 1u. Call a pair of vertices linked if they have an edge between them

and unlinked otherwise. Call a set of vertices independent if no two vertices in the set

share an edge.

Theorem 1.6 (Fouvry-Kluners sieve [25]). Fix Φ : V V Ñ F2, a ¡ |V |1, M P 4Z,

a subset T ppZMZqq|V |, and a multiplicative character χ : ppZMZqqV Ñ t1u.Then

¸±Du X

aωp±Duqµ2

¹Du

χppDuquPV q

¹u,vPV

Du

Dv

Φpu,vq

¸UγpUq

¸pn,Mq1,n X

µ2pnq pa|U |qωpnq OXplogXqa|U |a1ε ,

The first sum is over Du P Z, pDu mod Mq P T , the objects U are independent sets

of GrpΦq of maximal size, and

γpUq φpMq|U |¸

phuqPT pUqχpphuquPV q

¹u,vPU

p1qphu12 qphv1

2 q

such that T pUq tt P T : tu 1 if u R Uu.

Fouvry and Kluners show that the following sum takes the form of the left-hand side

of the above expression: ¸|d| X

2rk4ClpQp?dqq

21

This follows from results due to Fueter [26], Redei and Reichardt [48] [49] [50].

Strictly speaking, Fouvry and Kluners only prove this result in the cases relevant

to the 4-rank, where M 4 or 8, a 12, V t0, 1, 2, 3uk, and specific choices of

B ppZMZqqV , Φ : V V Ñ F2, and χ : ppZMZqqV Ñ t1u. However the sieving

technique applies in more generality with relative ease. We will sketch the main points

of the sieving technique used to prove Theorem 1.6 and refer to [25] for their proofs.

Sketch of the Fouvry-Kluners Sieve. For convenience, we denote the left hand side of

Theorem 1.6 by

SpXq :¸

±Du X

aωp±Duqµ2

¹Du

χppDuquPV q

¹u,vPV

Du

Dv

Φpu,vq

Let ∆ 1 loga|V |X. Define A to be a tuple pAuq of variables with each Au

corresponding to Du, and each Au ∆j for some j ¥ 0. We can partition S pXqaccording to the various A, by letting S pX,Aq be the sum over pDuq with

±Du X

restricted to summands for which Au ¤ Du ¤ ∆Au. Hence

S pXq ¸A

S pX,Aq .

Note that if ∆ 1 loga|V |X then there are OplogXq|V |p1a|V |q

possible A

with S pX,Aq not empty. This is since there are OplogXqp1a|V |q

choices for each

1 Au ¤ X.

Let Ω ea|V | plog logX B0q, where e is Euler’s number and B0 is a constant

defined in Fouvry-Kluners [25]. Noting a ¡ |V |1

Lemma 1.7 ( [25]). Let R be the sum of the terms in SpXq which satisfy: at least one

Du has ω pDuq ¡ Ω. Then

R OX plogXq1

.

22

Lemma 1.8 ( [25]). Let F1 be the set of A which satisfy±

u ∆Au ¡ X. Then

¸APF1

S pX,Aq OX plogXq1

.

For the next two lemmas we will need to define

X; exp plogηXq

for some small constant η.

Define U to be the maximal size of an independent set U GrpΦq. Then

Lemma 1.9 ( [25]). Let F2 be the set of A satisfying: at most U vertices u satisfy

Au ¡ X;. Then ¸APF2

S pX,Aq ! X plogXqapU1η|V |q1

Lemma 1.10 ( [25]). Let F3 be the set of A satisfying: there exist two linked indices u

and v with Au ¥ X; and Av ¥ 2. Then

¸APF3

S pX,Aq OX plogXq1

.

Consider the set of A satisfying

A is not in Fi for any i 0, 1, 2, 3. (1.1)

Combining the above lemmas we reduce our expression to

S pXq ¸1

A

S pX,Aq OX plogXqaUa1ε

where the sum is over A satisfying p1.1q, after taking η εa1|V |1.

Note this condition implies that there are at least U variables Au ¡ X; and they

are all pairwise unlinked. Call the set of such vertices U . U is the size of a maximal

23

independent set of GrpΦq, so U tu P V : Au ¡ X;u. Moreover, for any v R U , the

maximality implies there exists a u P U linked to u. By removing the family F3, this

implies Av 2, so that Dv ∆Av 2 for sufficiently large X.

Two vertices are unlinked if and only if Φpu,vq Φpv,uq. Therefore, the nontrivial

terms in SpX,Aq for A corresponding to the maximal independent set U GrpΦq are

given by

SpX,Aq ¸

±Du X

aωp±Duqµ2

¹Du

χppDuquPV q

¹tu,vu

p1qΦpu,vqDu12

Dv12

OXplogXqaUa1ε ,

where the sum is over Du 1 whenever u R U and pDu mod Mq P T .

The final step is to separate Du from its equivalence class modulo M . We refer to

another lemma from the work of Fouvry-Kluners:

Lemma 1.11 ( [25]). For any fixed tuple phuq P ppZMZqqU when 4 |M ,

¸pDuhu mod Mq

µ2¹

Du

aωp

±Duq 1

φpMqU¸

±Du X

µ2

¹p|M

p¹

Du

aωp±Duq

OX plogXq1

,

where φpMq is the Euler totient function.

Applying Lemma 1.11 to SpX,Aq, it follows that

SpX,Aq φpMqU¸

phuqPT pUqχpphuquPV q

¹tu,vu

p1qΦpu,vqDu12

Dv12

¸pn,Mq1,n X

paUqωpnqµ2pnq

OXplogXqaUa1ε

γpUq¸

pn,Mq1,n XpaUqωpnqµ2pnq O

XplogXqaUa1ε

24

Taking a sum over the remaining A, which correspond to independent sets U GrpΦqof maximal size, concludes the sketch of this sieve.

25

Chapter 2

Nonabelian Cohen-Lenstra

Moments

This chapter presents the author’s work towards verifying cases of Wood’s conjecture,

also referred to as nonabelian Cohen-Lenstra moments.

Conjecture (Wood’s Conjecture [59]). For any admissible pair pG,G1q Let c be the

number of conjugacy classes of elements g P G1G of order 2. If pG,G1q is “good”, i.e.

c 1, then

EpG,G1q |H2pG1, cqr2s||AutG1pGq| and EpG,G1q |H2pG1, cqr2s|

|c||AutG1pGq|

If pG,G1q is “not good”, i.e. c ¡ 1, then EpG,G1q 8.

A majority of the results pertain to “not good” pairs. As demonstrated by the em-

bedding problem discussed in Section 1.2.1, a lot is known about the Galois cohomology

of nilpotent extensions. Many of the cases for which we prove Wood’s conjecture, or

begin the process of proving Wood’s conjecture, are cases for which G1 is nilpotent.

This allows us to take advantage of the nice Galois cohomological properties of nilpo-

tent extensions. It turns out that if G1 is nilpotent, then any admissible pair pG,G1q is

necessarily “not good”.

26

We do not stop at EpG,G1q 8, but also consider the main term of the function

NpG,G1;Xq. The cohomological and analytic methods used in this chapter suggest

that proving an asymptotic counting result is attainable in many cases described in

Section 2.3 by using similar methods to those used for NpHk8 , H

k8 C2;Xq in Section

2.2.2.

2.1 Admissible Pairs

An admissible pair is defined by Wood [59] to be a pair of groups pG,G1q satisfying

the following:

• G1 G o C2 pGGq C2 with quotient map π : G o C2 Ñ C2,

• pp1, 1q, σq P G1 where C2 xσy,

• G1 X kerπ ¤ GG surjects onto the first coordinate,

• G1 is generated by elements of order 2 not in ker π.

If there exists a pG,G1q-extension LQ, i.e. extensions LKQ such that LK is

unramified, GalpLKq G, and GalprLQq G1, then pG,G1q is an admissible pair.

2.1.1 GI-extensions

Given an unramified extension KQp?dq normal over Q, GalpKQq is generated by its

inertia subgroups all of which necessarily have order 1 or 2. Moreover, the nontrivial

inertia groups of order 2 are not contained in GalpKQp?dqq. This motivated Boston

to make the following definition [10]:

27

Definition 2.1. Given G G1 of index 2, we say G ãÑ G1 is a GI-extension of G if G1

is generated by involutions (elements g P G1 with g2 1) not contained in G.

Here the GI can be taken to stand for “Generated by Involutions”. A pair pG,G1qwith rG1 : Gs 2 is admissible if and only if G1 is a GI-extension of G. Boston gives an

equivalent formulation of this definition [10]:

Lemma 2.2. G1 is a GI-extension of G iff G1 G C2 where C2 acts on G by an

automorphism σ P AutpGq such that G is generated by tg P G : gσ g1u

Notice that GI can coincidentally be taken to stand for “Generator Inverting”. As

such, we call any σ satisfying the above condition a GI-automorphism of G. Conse-

quently [10]:

Corollary 2.3. There is a bijection between GI-extensions of G up to isomorphism

and tC OutpGq : C is a conjugacy class containing the coset of a GI-automorphism

σ of Gu.

Here OutpGq AutpGqInnpGq denotes the group of automorphisms modulo conju-

gations and an isomorphism of GI-extensions G1 and G2 of G is defined to be a group

isomorphism φ : G1 Ñ G2 such that φpGq G.

Consider the following examples:

Lemma 2.4. 1. If A is an abelian group, then A has a unique GI-extension given by

the automorphism σpaq a.

2. G C2 is a GI-extension of G iff G is generated by elements of order 2.

3. Sn is a GI-extension of An.

28

Proof. 1. Suppose A has a GI-extension G1 A C2. Then by lemma 2.2 A is

generated by elements ta P A : aσ au for σ P C2 the generator. A is abelian,

so this implies aσ a for all a P A.

2. G C2 is a GI-extension of G iff G is generated by tg P G : gσ g1u for σ P C2

a generator by lemma 2.2. But gσ g in GC2, so tg P G : gσ g1u tg P G :

g2 1u.

3. An is a normal subgroup of index 2 in Sn. Sn is generated by transpositions, which

are elements of order 2 not contained in An, making it a GI-extension.

Corollary 2.5. There exist groups G with more than one non-isomorphic GI-extension

This follows from points 2 and 3 in the above lemma, as An is generated by elements

of order 2 for n ¥ 5 and Sn An C2.

In generalizing the Cohen-Lenstra heuristics to nonabelian groups, it is then more use-

ful to divide the question into cases based on the isomorphism class of the GI-extension

GalpKQq of GalpKQp?dqq to account for the differences in the action. Consider the

following generalization made by Bhargava [6]: the expected number of times the pair

pG,G1q with G ¤ G1 occurs as pGalpKQp?dq, GalpKQqq where G1 is a GI-extension

of G as d varies. Define:

EpG,G1q : limXÑ8

°kPD

X#tKk unramified with Galois group G, GalpKQq G1u°

kPDX

1

Note that this does not alter the Cohen-Lenstra moments for abelian groups, as all

abelian groups have a unique GI-extension. When expressed in this form, Bhargava

29

proved the following for n 3, 4, 5:

EpSn, Sn C2q 8,

EpAn, Snq 1

n!,

EpAn, Snq 1

2pn 2q! .

In general, one has EpGq °G1 EpG,G1q summed over possible GI-extensions of G,

so solving for EpG,G1q will give us EpGq. As a consequence, if G does not have any

GI-extensions then there cannot exist any unramified extensions LQp?dq Galois over

Q with GalpLQp?dqq G. Therefore the numerator of EpGq is identically 0 for all

X, forcing EpGq 0.

Corollary 2.6. For primes p 2, infinitely many finite p-groups G do not have a

GI-extension. In particular, EpGq 0.

Moreover, this is true of “asymptotically almost all” (in the sense of [31]) finite

p-groups.

Proof. A consequence of Horosevskii [32] shows that infinitely many p-groups have au-

tomorphism group also a p-group (see a Section 5.2 of a survey paper by Helleloid [30]

for a discussion of this consequence). A GI-automorphism necessarily has order dividing

2, so for p 2 infinitely many finite p-groups have at most one such automorphism, the

identity. This is a GI-automorphism iff the group is generated by elements of order 2,

which is not the case for p 2.

Moreover, [31] show that the number of p-groups G with AutpGq also a p-group

is “asymptotically almost all p-groups” when p-groups with a fixed lower p-length are

ordered by the number of generators, or when p-groups with a fixed number of generators

30

are order by the lower p-length. This suggests that having a GI-extension may be a rare

among among certain families of finite groups, see [31] for more details.

2.1.2 A family of groups without GI-extensions

Definition 2.7. Let q pn be a prime power and d | q 1, and define Gpq, dq tx ÞÑax b : a, b P Fq with ad 1u. Equivalently, Gpq, dq Cn

p Cd where Cnp is the additive

group of Fq, and Cd ¤ Fq acts on it by multiplication.

The goal of this section will be to determine how many GI-extensions Gpq, dq has

for each choice of q and d. Our strategy will be to realize the automorphism group of

Gpq, dq as a matrix group, and use this to determine what form a GI-automorphism can

take as a matrix acting on Gpq, dq.The action of Gpq, dq on Fq makes it a Frobenius group (see p. 252 of [51] or [57]):

Definition 2.8. A group G is a Frobenius group if there is an action of G on some

set X such that every nonidentity element has at most one fixed point and at least one

nonidentity element has a fixed point. Then the collection of elements with no fixed

points together with the identity form a normal subgroup called the Frobenius kernel K,

G splits over K, and any complement to K in G is called a Frobenius complement.

G K H for GK H.

In the case of Gpq, dq, K Cnp and H Cd. We consider this viewpoint in order to

take advantage of a recent result due to Wang [57]:

Corollary 2.9. If G K φ H is a Frobenius group with abelian Frobenius kernel K

and action φ : H Ñ AutpKq, then AutpGq K NAutpKqpφpHqq ¤ HolpKq.

31

Here we take the holomorph HolpKq to denote K AutpKq with the natural action.

Lemma 2.2 in Wang’s paper takes advantage of the fact that G has trivial center, i.e.

ZpGq 1, so that G may be identified with InnpGq ¤ AutpGq the automorphisms by

conjugation. Using this identification, he views K as a subgroup of AutpGq and shows

all automorphisms of G are determined by conjugation by an element of K and another

automorphism of K.

We then necessarily have that AutpGpq, dqq embeds in HolpCnp q. In particular we can

realize these as matrix groups:

Lemma 2.10. Let xd P Fq be an element of multiplicative order d. xd acts on the

additive group Fq Cnp by multiplication, so choose a basis for Cn

p and identify xd with

the matrix Xd P GLnpFpq AutpCnp q defined by Xdv xdv. Then

HolpCnp q

$'&'%A b

0 1

P GLn1pFpq : A P GLnpFpq, b P Fnp

,/./-Gpq, dq

$'&'%Xk

d b

0 1

P GLn1pFpq : 0 ¤ k d, b P Fnp

,/./-are groups of block upper triangular matrices, with an n n and a 1 1 block on the

diagonal.

Using this lemma and Corollary 2.9, we can show that any automorphism of Gpq, dqcan be viewed as a matrix in HolpCn

p q acting on Gpq, dq ¤ HolpCnp q by conjugation.

Applying the definition of GI-automorphism to matrix operations gives the following:

Lemma 2.11. Fix a matrix p T a0 1 q P HolpCn

p q of order 1 or 2 (i.e. such that T 2 1 and

Ta a). p T a0 1 q is a GI-automorphism of Gpq, dq iff Gpq, dq is generated by elements

of the formXk

d b0 1

with TXk

dT Xkd and TXk

da Tb a Xkd b.

32

We conclude this section with a complete classification of GI-extensions of Gpq, dq,first separating out those without a GI-extension and then counting the number of GI-

extensions for the remaining groups.

Theorem 2.12 (A. [1]). Gpq, dq has a GI-automorphism iff Dl such that pl 1 mod d

for q pn.

Proof. If d 1, then p 1 mod 1 trivially. Morevoer, Gpq, 1q Cnp is abelian, which

has a unique GI-automorphism sending x ÞÑ x. For the rest of this proof, suppose

d ¡ 1.

pñq We then necessarily have TXkdT Xk

d for a generating set of powers of Xd in

xXdy (and so for every power of Xd). T acts by conjugation on the matrix algebra

FprXds GLnpFpq Y t0u, so without loss of generality T acts on Fppxdq which is iso-

morphic to FprXds as an algebra. Conjugation of matrices is a ring automorphism, so T

acts on Fppxdq by some power of Frobenius φp. Thus x1d xTd φlppxdq xp

l

d for some

l, so that pl 1 mod d.

pðq Fix m rFppxdq : Fps. The power of Frobenius φ`p fixes Fp and maps xd ÞÑ x1d by

p` 1 mod d. Let α φ`p be the involuting automorphism of Fppxdq. α acts on the

additive group of Fppxdq, which is isomorphic to Fmp as an m-dimensional vector space.

Fix a basis of Fppxdq (such as 1, xd, x2d, ..., x

m1d ) to identify with a vector space basis of

Fmp and let T be the matrix in GLmpFpq given by the action of α on Fmp through this

identification. (Note: if d 2 then T is the identity matrix. Otherwise T has order 2.)

33

First, suppose Fq Fppxdq i.e. n m. Consider the matrix p T 00 1 q acting on Gpq, dq

by conjugation, we will show that conjugation by this matrix is a GI-automorphism of

Gpq, dq. It suffices to show (by lemma 2.11) that Gpq, dq is generated by elements of the

formXk

d b0 1

such that TXk

dT Xkd and Tb Xk

d b. The first equality holds by

construction, as Txd xp`

d x1d . The second holds for b 0 and b xkd 1. These

elements generate the following:Xkd xkd 1

0 1

Xk

d 0

0 1

1 xkd 1

0 1

.To show that these matrices, with

Xk

d 00 1

, generate Gpq, dq it suffices to show that xkd1

for k 1, 2, ..., d1 span Fppxdq Fq. But notice d °d1k1pxkd1q and gcdpd, pq 1.

Thus 1 P Fp is contained in the span, and consequently so is xkd for any k. These are a

basis generating Fppxdq Fq, concluding the proof of this case.

Now suppose rFq : Fppxdqs ¡ 1 and keep T as above. Extend a basis tviu of FqFppxdqby a basis twju of FppxdqFp to a basis twjviu of FqFp ordered lexicographically. DefinerT P GLnpFpq to be a block diagonal matrix with T ’s along the diagonal and consider rT 0

0 1

. We will show that conjugation by this matrix is a GI-automorphism of Gpq, dq.

All elements of the formXk

d bvi0 1

satisfy the equation in lemma 2.11 whenever

Xk

d b0 1

does in the first case by rTwjvi pTwjqvi. The first case showed that such values of

b generate Fppxdq, so it follows that bvi generate Fq and that the matricesXk

d bvi0 1

satisfying the equations in lemma 2.11 generate Gpq, dq.

Theorem 2.13 (A. [1]). Gpq, dq has at most one GI-extension.

Retain all notation from the previous theorem.

34

Proof. It suffices to count GI-automorphisms up to inner automorphism and conjugation.

We know that AutpGpq, dqq Cnp NAutpCn

p qpCdq ¤ HolpCnp q. Fix a basis twjviu for

the extension FqFp as in the previous theorem, where tviu is a basis for FqFppxdq and

twju is a basis for FppxdqFp. Then automorphisms of Gpq, dq are represented by matri-

ces p T a0 1 q with a P Fq and T P NGLnpFpqpCdq : NpCdq.

Notice how any matrix S P NpCdq must satisfy Spxyq SpxqSpyq for any x P Fppxdqand y P Fq. Thus S|Fppxdq P AutpFppxdqq i.e. is some power of Frobenius φmp . Define

PS to be the linear map sending wjvi ÞÑ wjSpviq. Then SP1S is the block diagonal

matrix sending wjvi ÞÑ φmp pwjqvi. Note that PS is an Fppxdq-linear map, showing that

NpCdq ApCdqBpCdq where ApCdq GalpFppxdqFpq is the group of block diagonal

powers of Frobenius and BpCdq GLpFqFppxdqq is the group of Fppxdq-linear bijections

Fq Ñ Fq. As a matter of fact, BpCdq is a normal subgroup giving NpCdq BpCdqApCdqwhere elements of ApCdq act on BpCdq by applying the corresponding power of Frobe-

nius to the coordinates of a matrix in BpCdq. Defining and showing BpCdq is a normal

subgroup can be done basis free, where choosing a section for the semidirect product is

then equivalent to the choice of basis tviu.

Up to inner automorphism, notice that any automorphism satisfies

p T a0 1 q p T 0

0 1 qI T1a0 1

So it suffices to consider automorphisms where a 0.

35

Given a GI-automorphism σ p T a0 1 q, we must have T φlp mod BpCdq where pl 1

mod d. Up to inner automorphism we have T defined up to being multiplied by Xkd .

Up to conjugation we have SXkdTS

1 Xpmkd STS1 where S φmp mod BpCdq. This

shows the two relations commute (as p is invertible mod d). It then suffices to count

equivalence classes of GI-automorphisms under the composite relation.

Now consider the set

MT !b : Dk s.t.

Xk

d b0 1

is inverted by σ

)If T comes from a GI-automorphism this must contain an FqFppxdq basis, in other words

there exists a basis such that T Bφlp under the semidirect product decomposition given

above such that B is diagonalizable with eigenvalues xkd for some values of k. Instead

of choosing a basis, this is equivalent to choosing a change of basis matrix P , such that

T decomposes into T pBP φlpP1qpPφlpP1q where BP φlpP1 is diagonalizable with

eigenvalues xkd for some values of k. Here, PφlpP1 is the block diagonal Frobenius

map in the new basis. So, in order to classify σ up to inner automorphism and up to

conjugation it suffices to count equivalence classes of the sequence tkiu of eigenvalue

exponents of B up to the following relations:

• ki k1i λ mod d for all i, given by inner automorphisms

• ki pmk1i mod d for all i, given by conjugation

• Changing B to BP φlpP1, by choosing a different basis

The first two relations follow from the above computations and are realatively easy to

36

work with. As for the last relation, a generalization of Hilbert’s Theorem 90 (see p.150-

151 of [53]) shows that every matrix U with Uφlp U1 is of the form P φlpP1. Suppose

B has eigenvalues xkid with a corresponding basis of eigenvectors vi. Then define U to

be the matrix such that Uvi xmid vi. Clearly, Uφlp U1, so there is a change of basis

matrix P such that T BUpPφlpP1q, and BU is a matrix with eigenvalues xkimid

by construction. Thus up to equivalence there is only one such sequence tkiu, implying

there is exactly one GI-extension.

2.2 Quaternion Group

There are some cases for which unramified extensions are easier to describe. Consider

the following result due to Lemmermeyer (paraphrased by combining the statements of

Propositions 4,5,6 and Theorem 1 in [40]):

Theorem ( [40]). Let k be a quadratic number field with discriminant d. There exists an

unramified extension Mk with GalpMkq H8, the quaternion group, which is normal

over Q if and only if

(a) GalpMQq D4 O C4,

(b) there is a factorization d d1d2d3 into three discriminants (called an H8-factorization),

at most one of which is negative,

(c) for all primes pi | di we haved1d2

p3

d1d3

p2

d2d3

p1

1.

Moreover, for each such H8-factorization d d1d2d3 there are exactly 2ωpdq3 such ex-

tensions Mk.

37

D4OC4 is notation used by Lemmermeyer to denote the direct sum of D4`C4 with

the center ZpD4q and the unique subgroup C2 ¤ C4 identified. If we write D4 xa, b :

a4 b2 bab1a 1y and C4 xc : c4 1y, then

D4 O C4 xa, b, c : a4 b2 c4 bab1a aca1c1 bcb1c1 c2a2 1y

We can express Lemmermeyer’s classification in a single formula. If kQ is a quadratic

field of discriminant d for some positive integer d P Z, then the number of unramified

H8-extensions Mk, Galois over Q, corresponding to the factorization d pd1qd2d3

is given by δapd1, d2, d3q, where

apd1, d2, d3q 1

8δ

¹p|d

1

d1d2

p

1

d1d3

p

1

d2d3

p

(2.1)

and where δ accounts for symmetry in the factorization, and is equal to 2 if d 0

and 6 if d ¡ 0. The number of unramified H8-extensions Mk, Galois over Q, is then

given by ¸dd1d2d3

apd1, d2, d3q,

where the sum is over factorizations with d1, d2, and d3 quadratic discriminants (so

that the sign of d agrees with the choice of ).

2.2.1 The Expected Number

Lemmermeyer’s classification can be used to compute the expected number of unramified

H8-extensions of imaginary/real quadratic fields. This will agree with Wood’s prediction

[59]:

Theorem 2.14 (A. [1]). EpH8q 8

38

We will detail the proof as found in section 4 of [1]. Lemmermeyer’s work tells us

that for some positive constant A

EpH8q A limXÑ8

1

X

¸d1d2d3 X

apd1, d2, d3q

The big idea from this section is to analyze the Dirichlet series¸apd1, d2, d3qpd1d2d3qs

and use a Tauberian theorem to determine the asymptotic behavior of¸d1d2d3 X

apd1, d2, d3q.

In fact, in order to show EpH8q 8 we only need a lower bound for

1

X

¸d1d2d3 X

apd1, d2, d3q

which tends to 8 as X Ñ 8. We will obtain such a lower bound by only considering

certain values for d. For any two odd discriminants d1 and d2, define the set

Dpd1, d2q tm : m is an odd quadratic discriminant and pm, d1d2q 1u

We will describe the asymptotic behavior of¸mPDpd1,d2q,d1d2m X

apd1, d2,mq

as X Ñ 8 by examining the corresponding Dirichlet series and applying a Tauberian

theorem.

Lemma 2.15. Let Dps, d1, d2q °mPDpd1,d2q a

pd1, d2,mqpd1d2mqs, then

Dps, d1, d2q pd1d2qs 1

8δ

¸a|d1

¸b|d2

¸mPA

d1d2

2ωpmqd1m

a

d2m

b

p1 χ4pmqqms,

where An tm P N : @ primes q | m, q

n

1 and m odd squarefreeu. Moreover

Dps, d1, d2q is holomorphic for Repsq ¡ 1.

39

Proof. The decomposition follows immediately from setting d d1d2m and d3 m in

the expression for apd1, d2, d3q.We will show here that the series converges absolutely for <psq ¡ 1. Indeed, we have

an upper bound of

pd1d2qs¸a|d1

¸b|d2

8

m1

2ωpmqp2qms 2pd1d2qs¸a|d1

¸b|d2

1

ζpsqζp2sq ,

which converges absolutely for <psq ¡ 1 by pd1d2qs, ζpsq, and ζp2sq converging abso-

lutely for <psq ¡ 1.

The terms outside the summations, pd1d2qs 18δ

, are holomorphic and zero-free on all

of C, and so may essentially be ignored when determining the existence and orders of

poles. We will deal with each summand on the right as follows:

Definition 2.16. Given a Dirichlet character χ, define

Mn ps, χq

¸mPA

n

2ωpmqχpmqms

This is defined so that we have the following:

Dps, d1, d2q pd1d2qs 1

8δ

¸a|d1

¸b|d2

d1

a

d2

b

M

d1d2

s, ab

M

d1d2

s, χ4

ab

We will show that M

n ps, χq can be meromorphically continued to an open neigh-

borhood of ts P C : Repsq ¥ 1u which has one simple pole at s 1 if χ is the trivial

character and is holomorphic otherwise. This will tell us that Dps, d1, d2q has only

one simple pole in this neighborhood at s 1 and what its residue is. From here,

Dps, d1, d2q °mPDpd1,d2q a

pd1, d2,mqpd1d2mqs is a Dirichlet series with positive co-

efficients, converges for Repsq ¡ 1, and can be meromorphically continued to an open

40

neighborhood of ts P C : Repsq ¥ 1u with one simple pole at s 1. A standard

Tauberian theorem (such as Delange’s Tauberian theorem [16]) then implies

¸mPDpd1,d2q,d1d2m X

apd1, d2,mq Ress1Dps, d1, d2qX.

We begin with the following properties for Mn ps, χq:

Lemma 2.17. Mn ps, χq satisfies the following properties:

• It is holomorphic for Repsq ¡ 1

• Mn ps, χq

¹p qnq1

1 2χpqqqs

• Mn ps, χqM

n ps, χq ¹q|np1 2χpqqqsq1

¸msqf

2ωpmqχpmqms

• Mn ps, χq

Mn ps, χq

¹

p qnq1

p1 4χ2pqqq2sq1¹q

1 2χpqq

qn

qs

• Mn ps, χq

Mn ps, χq

¹p qnq1

p1 4χ2pqqq2sq1¹q

1 2χpqq

qn

qs

where each product is restricted to q 2 and m sqf means the sum is over odd

squarefree values of m.

The proofs of these follow by computation of their Euler products. We will show that

Mn is both zero and pole free on some domain containing Repsq ¥ 1 with a possible

exception at s 1, but first we must prove the following lemma:

Lemma 2.18. Consider the series given by Euler product±

p-b p1 bχppqpsq for b a

nonzero integer. This series is meromorphic on the zero-free region of Lps, χq, whose

only pole or zero is s 1 of order b if χ 1. It is holomorphic on the zero-free region

of Lps, χq if χ 1.

41

Proof.

¹p-b

1 bχppqps ¹

p-b

p1 bχppqpsq

1°|b|k1

|b|k

psgnpbqqkχppqkpksp1 sgnpbqχppqpsq|b|

This identity is achieved by multiplying top and bottom by p1 sgnpbqχppqpsq|b|,where sgnpbq is the sign of b. In particular, the numerator has no term which is linear

in ps. Thus a computation with natural logs shows the numerator is holomorphic on

Repsq ¡ 12. Denote the numerator by Gpsq. Then

¹p-b

1 bχppqps

$''&''%Gpsq±

p|bp1χppqpsqbLps, χqb b ¡ 0

Gpsq±p|bp1χppqpsqb

Lps,χqLp2s,χ2q

bb 0

The result is clear from this decompostion and the fact that Lps, χq is holomorphic on

C if χ 1 and meromorphic with a unique simple pole at s 1 if χ 1.

Proposition 2.19. Mn ps, χq is meromorphic on the intersection of the zero-free regions

of Lps, χq and Ls, χ

n

. M

n ps, χq has only one pole or zero at s 1 of order 1

iff χ or χ n

is trivial, and is holomorphic and zero-free otherwise. Moreover it follows

that

Ress1Mn ps, 1q

d¹p

p1 2p1qp1 p1q2

1 2pn

p1

1

pn

p1

2

3p2 2n

q4

d¹q|np1 2q1q1

¹p qnq1

p1 4q2q1L

1, n

Ress1Mn

s, n

d¹p

p1 2p1qp1 p1q2

1 2pn

p1

1

pn

p1

2

3p2 2n

q4

d ¹p qnq1

p1 4q2q1L

1, n

Each product is over odd primes p and q satisfying the given conditions.

42

Proof. Using the previous lemma, we can conclude that both Mn ps,χq

Mn ps,χq and M

n ps,χqM

n ps,χq are

meromorphic on the zero-free region of Ls, χ

n

. Additionally, they each have only

one pole lying at s 1 of order 2 and 2 respectively iff χ n

1, and are holomor-

phic otherwise. As in the previous lemma, we let Gpsq ±pp1 2χppq p

n

psqp1

χppq pn

psq2 which is holomorphic on Repsq ¡ 1

2. Moreover, because of our knowledge

of the reciprocal, we know Gpsq is zero-free on this region and we have the following:

Mn ps, χq

Mn ps, χq

Gpsqp1 χp2q2sq2

¹p qnq1

p1 4χ2pqqq2sq1Ls, χ

n

2

We handle the reciprocal similarly, implying all the components are holomorphic and

zero-free except possibly the L-function.

In addition, we can also use the previous lemma to show Mn ps, χqM

n ps, χq is mero-

morphic on the zero-free region of Lps, χq with a pole of order 2 iff χ 1, and holo-

morphic otherwise. Let F psq ±pp1 2χppqpsqp1 χppqp2q2, similarly shown to be

holomorphic and zero-free on the region in question, showing that

Mn ps, χqM

n ps, χq F psq

p1 χp2q2sq2

¹q|np1 2χpqqqsq1Lps, χq2

where each component is holomorphic and zero-free on the region in question except

possibly the L-function.

Multiplying these two together gives:

Mn ps, χq2 GpsqF psq

¹q|np1 2χpqqqsq1

¹p qnq1

p1 4χ2pqqq2sq1

1 χp2q

2

n

2s

2 1 χp2q2s2

Lps, χq2Ls, χ

n

2

43

This function is meromorphic in the intersection of the zero-free regions of Lps, χq and

Ls, χ

n

, and whose only pole comes from one of the L-functions. In particular, since

every component is zero-free on this region (save a possible pole at s 1), we take a

branch cut along the negative real axis with starting point s 1 and take the square

root of both sides of this equation. This shows that

Mn ps, χq

dGpsqF psq

¹q|np1 2χpqqqsq1

¹p qnq1

p1 4χ2pqqq2sq1

1 χp2q

2

n

2s

1 1 χp2q2s1

Lps, χqLs, χ

n

where the depends on the root chosen for the other functions. This is meromorphic on

the region in question, whose only possible pole is at s 1 coming from an L-function.

Calculation of the residues is can then be done by retracing through the definitions of

Gpsq and F psq.

Going back to the series in question, we can conclude the following:

Corollary 2.20. Dps, d1, d2q is meromorphic on a finite intersection of zero-free re-

gions of L-functions, which has one simple pole at s 1 of residue

1

8δd1d2

3

2

2d1d2

4

L1,

d1d2

d ¹q|d1d2

p1 2q1q1 p1q d112

d212

gffe¹

p

p1 2p1qp1 p1q2

1 2

p

d1d2

p1

1

p

d1d2

p1

2

gffe ¹

q

d1d2

1

p1 4q2q1

Each product is over odd values of p and q.

44

Proof. The residues only come from two of the terms in the sum expressing Dps, d1, d2q,namely those terms corresponding to a b 1 and a d1, b d2. We can choose our

roots in such a way that the can factor out. This being a simple pole is a consequence

of the residue being nonzero. Most of it was concluded to be nonzero in the preceding

proof, and the remaining bits

pd1d2q1 1

8δ

d ¹q|d1d2

p1 2q1q1 p1q d112

d212

are trivially nonzero.

As stated above, the Tauberian theorem shows that°mPDpd1,d2q,d1d2m X a

pd1, d2,mq Ress1Dps, d1, d2qX. Note, however, that this is a sum of all positive terms, so the

residue itself must be positive. We can make the following bounds:gffe¹p

p1 2p1qp1 p1q2

1 2

p

d1d2

p1

1

p

d1d2

p1

2

gffe ¹

q

d1d2

1

p1 4q2q1

gfffe¹

p

p1 2p1qp1 p1q2

1 p2 2

p

d1d2

p3

¹

qd1d2

1

p1 4q2q1

¥d¹

p

p1 2p1qp1 p1q2 p1 p2 2p3q

This is a constant independent of d1, d2. Because d pd1qd2m factors into a

product of discriminants, we know d1 1 mod 4 and d2 1 mod 4. Then, noting

the need for every term to be positive, it follows that

45

1 p1q d112

d212

d ¹q|d1d2

p1 2q1q1 ¥ 1

This is also a constant independent of d1, d2. Lastly, we can bound3p2

2d1d2

q

4

¥ 3

4

We conclude:

Lemma 2.21. There exists a positive constant c independent of d1, d2 such that

Ress1Dps, d1, d2q ¥ cL

1,

d1d2

d1d2

To conclude this section, it suffices to combine the families Dpd1, d2q as d1 and d2

vary.

Corollary 2.22. EpH8q 8

Proof. We only need a lower bound on the expected number to be infinite, so let us only

consider odd discriminants d. By definition, it follows that

EpH8q A limXÑ8

1

X

¸d X

¸dd1,d2,d3

apd1, d2, d3q

¥ limXÑ8

1

X

¸d1,d2 N

¸mPDpd1,d2q,d1d2m X

apd1, d2,mq

¥¸

d1,d2 NlimXÑ8

1

X

¸mPDpd1,d2q,d1d2m X

apd1, d2,mq

¸

d1,d2 NRess1Dps, d1, d2q

¥¸

d1,d2 NcL

1,

d1d2

d1d2

¥ c

4

¸d N2

L1, d

d

46

Here we take N ¡ 0 any positive integer and the sums are over d, d1, and d2

odd quadratic discriminants. Obviously this is a very weak lower bound, given all the

information we have dropped, but it is sufficient. A result of Goldfeld-Hoffstein (Theorem

2 in [28]) states that°m Lpw, χmq|m|s has a pole at s 1 for Repwq ¥ 12, which

implies the sum diverges as we take N Ñ 8.

2.2.2 The Asymptotic Count

Proving that EpH8q 8 leaves something to be desired. Fouvry’s and Kluners’

average value of the 4-rank [25] does more than merely prove that EpC4q 8, they

verify Gerth’s heuristic and give an exact distribution. Rephrasing their work in the

terms used in this paper, they prove

NpCk4 , C

k4 C2;Xq ckXplogXq2k1,

where ck is the kth moment of the 4-rank of the class group given in their paper. This

result implies that EpCk4 q 8 as we expect from Wood’s conjecture [59], but also gives

much more information about how many such extensions exist. A result of this form

answers the question “How big is this infinity?” For example, EpC4q EpH8q 8,

but this does not mean that the number of unramified extensions of quadratic fields

with these Galois groups are roughly the same. There are asymptotically more for H8,

which we see from the following theorem:

Theorem 2.23 (A.-Klys [4]). Let k P Z¥1, G Hk8 , and G1 Hk

8 σ C2. Define

Surjσ pGal pKurKq , Gq be the set of surjections which lift to a surjection Gal pKurQq Ñ

47

G1. Then

¸dPD

X

|Surjσ pGal pKurKq , Gq|

1

4

k

¸dPD

X

3kωpdq

OX plogXq3k2ε

and

¸dPD

X

|Surjσ pGal pKurKq , Gq|

1

24

k

¸dPD

X

3kωpdq

OX plogXq3k2ε

.

In particular, this tells us that

NpHk8 , H

k8 C2;Xq rckXplogXq3k1,

for an explicit constant rck . Wood [59] gives some results in the function field setting

which suggest that this asymptotic, and the one for Ck4 , are not surprising.

The strategy for proving Theorem 2.23 is to adapt the proof given by Fouvry-Kluners

in [25]. Lemmermeyer’s classification of unramified H8-extensions of quadratic fields is

enough to make this work. We restated the Fouvry-Kluners sieve in Theorem 1.6 in the

form that will be most convenient for us (see Section 1.2.2 for more details and a sketch

of the proof). The proof is neatly separated into three parts: (1) the set-up, where

we determine which values V , Φ, a, T , M , and χ from the hypothesis of Theorem 1.6

correspond to our problem, (2) computing the independent sets U GrpΦq of maximal

size, and (3) computing the constant term°γpUq (in particular showing it is nonzero).

Set-up

It will be convenient to divide the quadratic disciminants into six families: d 0 with

d 1 mod 4, d 0, 4 mod 8 and d ¡ 0 with d 1 mod 4, d 0, 4 mod 8. These

families are qualitatively similar, and they will only differ by the value M and the

48

character χ. In the main body of this dissertation we will address the case d 0 and

d 1 mod 4, while the rest of the cases will be covered in the appendix.

If we include degenerate factorizations in Lemmermeyer’s classification of unramified

H8-extensions, then it follows for a quadratic field K of discriminant d

|HomσpGalpKurKq, H8q| ¸

dd1d2d3

8apd1, d2, d3q,

where the sum is over factorizations into odd, squarefree integers with d1 1 mod 4

and d2, d3 1 mod 4 (Note the factor of 8 is to account for automorphisms of H8 which

lift to G1. |HomσpGalpKurKq, H8q| is expanded as follows:

¸dd1d2d3

1

2

¹p|d

1

d1d2

p

1

d1d3

p

1

d2d3

p

1

2

¸dd1d2d3

¸a|d3

d1d2

a

¸b|d2

d1d3

b

¸c|d1

d2d3

c

1

2

¸dd1d2d3

¸d3D1D4

D0D3D2D5

D4

¸d2D2D5

D0D3D1D4

D2

¸d1D0D3

D1D4D2D5

D0

1

2

¸d±Di

1

D2D4

¹i,j

Di

Dj

Φpi,jq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3 1 mod 4 and D1D4, D2D5 1 mod 4. This is the exact set-up needed to

apply Theorem 1.6.

49

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| 1

2

¸d±Ei

1

E2

¹i,j

EiEj

Ψpi,jq,

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1 1 mod 4 and E2E3 1 mod 4.

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq1

We can put all of these together to give

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

1

2j1j2j32ωpdqj3

j1¹`1

¸d±D

p`qi

1

Dp`q2 D

p`q4

¹i,j

Dp`qi

Dp`qj

Φpi,jq

j1j2¹`j11

¸d±E

p`qi

1

Ep`q2

¹i,j

Ep`qi

Ep`qj

Ψpi,jq

50

We must get this back into the form in Theorem 1.6. Set Vj1,j2 t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand the expression as follows:

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

2ωpdqj3

2k

¸d±Du

χj1,j2ppDuquPV q¹u,v

Du

Dv

Φj1,j2pu,vq

,

where the character is given by

χj1,j2ppDuquPV q j1¹`1

1±

u`2,4Du

j2¹

`j11

1±u`2Eu

,

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du

$''&''%1 mod 4 1 ¤ ` ¤ j1, i 0, 3

1 mod 4 1 ¤ ` ¤ j1, i 1, 2, 4, 5

¹u`i,i1

Du

$''&''%1 mod 4 j1 1 ¤ ` ¤ j1 j2, i 0, 1

1 mod 4 j1 1 ¤ ` ¤ j1 j2, i 2, 3

For each partition k j1 j2 j3 we can now apply Theorem 1.6 with M 4 and

T ppZ4ZqqV describing the above congruence conditions to compute

¸d X,d1 mod 4

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

51

Independent Sets of Maximal Size

Call V1 t0, 1, 2, 3, 4, 5u and V2 t0, 1, 2, 3u so that Vj1,j2 V j11 V j2

2 . Recall that

we call two indices u,v linked if they share an edge in GrpΦj1,j2q, i.e. Φj1,j2pu,vq Φj1,j2pv,uq 1, and unlinked otherwise. When j1 0 the maximal unlinked subsets of

Vj1,j2 are determined by Fouvry-Kluners [25] and are of size 2j2 . We will now determine

the largest maximal unlinked sets when j2 0.

Proposition 2.24. Let A t1, 3, 5u and B t0, 2, 4u. Let S tA,Buk. The largest

maximal unlinked sets are all of size 3k and correspond bijectively to elements of S. The

set corresponding to s P S is

Us u P V k

1 | ui P si(.

To simplify notation we will refer to the largest maximal unlinked set corresponding

to s P S as being of type s or simply as a type.

Proof. Consider the graph Gk GrpΦk,0q with vertices t0, 1, 2, 3, 4, 5uk and adjacency

matrix given by rGks rBkpu,vqs mod 2, where we define Bkpu,vq Φkpu,vq Φkpv,uq. Unlinked sets are exactly the independent sets of Gk. Notice for k 1 that

G1 is a cyclic graph with 6 vertices, and has largest maximal independent sets given by

A and B. We use this as a base case for induction.

Suppose the proposition holds true for k 1, and let U Gk be independent, and

partition it into U ²5i0Ci where Ci tpu, iq P U : u P Gk1u. Call ci |Ci|, so that

we have |U | °ci. We know that

rGks rBkpu,vqs rBk1ppu1, ..., uk1q, pv1, ..., vk1qqs rB1puk, vkqs

52

The subgraph induced by U inside of Gk corresponds to a submatrix rU s 0 in rGksalong the vertices of U . In particular,

rBk1ppu1, ..., uk1q, pv1, ..., vk1qqs rB1pi, jqs

for all u P Ci, v P Cj. Ordering indices lexicographically, with the kth entry weighted

highest, it follows that

rBk1|U s

0c0c0 1c0c1 0c0c2 0c0c3 0c0c4 1c0c5

1c1c0 0c1c1 1c1c2 0c1c3 0c1c4 0c1c5

0c2c0 1c2c1 0c2c2 1c2c3 0c2c4 0c2c5

0c3c0 0c3c1 1c3c2 0c3c3 1c3c4 0c3c5

0c4c0 0c4c1 0c4c2 1c4c3 0c0c4 1c4c5

1c5c0 0c5c1 0c5c2 0c5c3 1c5c4 0c5c5

Where 0nm and 1nm are block matrices of dimension n m with all 0 and 1 entries

respectively.

Lemma 2.25.

ci1 ci1 ¤

$'''''''''''&'''''''''''%

2 3k1 ci ci2 ci4 0

3k1 ci 0, else

2p3k1 1q ci 0, ci2 ci4 0

3k1 1 ci 0, else

(2.2)

Proof. For any two disjoint sets of vertices V and W in Gk1 consider a complete bipar-

tite subgraph KV,W Gk1 on V and W . Since V is an unlinked set, by the inductive

hypothesis |V | ¤ 3k1 with equality if and only if V Ut for some t P S, and similarly

for W .

53

Suppose V H. Fix s P S. Then for any t P S and any v P V X Ut we have that

there exists a u P Us defined by

uj

$''&''%vj tj sj

vj3 tj sj

such that Bk1pu,vq 0, which implies u R W . Thus W Us and by the induction

hypothesis |W | ¤ 3k1 1. By symmetry the same is true for V if W H.

Define p : Gk Ñ Gk1 to be the projection forgetting the kth coordinate. Notice that

p|Ciis injective for all i values.

Suppose ci 0 ci2 ci4 0. Then we use the trivial bound: the submatrix on

vertices in Cj is a block zero matrix, so that ppCjq is an independent set of Gk1. Thus,

cj ¤ 3k1, so ci1 ci1 ¤ 2 3k1 by the inductive hypothesis.

Suppose ci 0 and without loss of generality ci2 0. Then for pu, i1q P Ci1 and

pv, i1q P Ci1, choose some pw, i2q P Ci2. We have Bk1pu,wq Bpi1, i2q 0

and Bk1pv,wq Bpi1, i2q 1, implying u v. Then we have ppCi1qXppCi1q H and ppCi1qYppCi1q is an independent set ofGk1, by Bk1pu,vq Bpi1, i1q 0.

Thus ci1 ci1 ¤ 3k1 by the inductive hypothesis.

Suppose ci 0, ci2 ci4 0. Then ppCi1q Y ppCiq is a complete bipartite

subgraph of Gk1, KV,W for |V | ci1 and |W | ci. Similarly for ci1, ci. Then

ci1, ci1 ¤ 3k1 1 and the result follows.

Suppose ci 0 and without loss of generality ci2 0. We can similarly prove

ppCi1q X ppCi1q H and ppCi1q Y ppCi1q Y ppCiq is a bipartite subgraph of Gk1,

KV,W with V ppCi1q Y ppCi1q and W ppCiq. So a combination of the previous

two results gives us this case.

54

Let I ti : ci 0u, we will separate cases based on the size of I.

If |I| ¥ 4 then we have

2|U | ¸i

ci1 ci1 ¸iRI

2ci

¤ 2p6 |I|q3k1

¤ 4 3k1

2 3k

so U is not of maximal size.

If |I| 3 we must separate into two cases. If I t0, 2, 4u or t1, 3, 5u then U ¤ 3k

with equality iff ppCjq is of maximum size for an independent set in Gk1 for all j R I.

But maximum implies it equals some Us, and I t0, 2, 4u or t1, 3, 5u implies we can

extend the type for k 1 to a type for k with U Us.

Otherwise, by symmetry we can assume I X t0, 2, 4u t0, 2u. Let j be the third

element of I. Then by the above lemma c1c3, c1c5 ¤ 3k1. Additionally c0c2 0 and

c0c4 c2c4 c4 ¤ 3k11 by at least one of c5, c3 nonzero. Lastly c3c5 ¤ 2 3k1.

Thus

2|U | ¸i

ci1 ci1

¤ 4 3k1 2p3k1 1q

¤ 2 3k 2

2 3k,

so U is not of maximal size.

If |I| 2 we need two cases. First, if I t0, 2, 4u or t1, 3, 5u then there exists

55

a j such that I tj 1, j 1u. Without loss of generality suppose j 1. Then

c0 c2 0 and c2 c4 c0 c4 ¤ 3k1 1. We also have c1 c3, c1 c5 ¤ 3k1 and

c3 c5 ¤ 2 p3k1 1q. Thus

2|U | ¸i

ci1 ci1

¤ 2 p3k1 1q 2 3k1 2p3k1 1q

¤ 2 3k 4

2 3k,

so U is not of maximal size.

Otherwise, I tj 1, j 1u. Then cj1 cj1 3k1 if j P I and 3k1 1 is j R I.

Thus

2|U | ¸i

ci1 ci1

¤ 2 3k1 4p3k1 1q

¤ 2 3k 4

2 3k,

so U is not of maximal size.

If |I| 1 suppose without loss of generality that 0 P I. Then c0 c2, c0 c4, c1 c5 ¤

56

3k1 and c1 c3, c2 c4, c3 c5 ¤ 3k1 1. Thus

2|U | ¸i

ci1 ci1

¤ 3k 3p3k1 1q

¤ 2 3k 3

2 3k,

so U is not of maximal size.

If |I| 0 then ci1 ci1 ¤ 3k1 1. Thus

2|U | ¸i

ci1 ci1

¤ 6p3k1 1q

¤ 2 3k 6

2 3k,

so U is not of maximal size.

We now combine these results to determine the largest maximal unlinked sets for all

j1, j2 ¡ 0.

Proposition 2.26. The largest maximal unlinked sets in Y are of the form V W where

V is a type in V j11 and W is a maximal unlinked set in V j2

2 . Thus the largest maximal

unlinked sets of Vj1,j2 are of size 3j12j2.

Proof. We fix j1 ¡ 0 and prove this by induction on j2. Let Gj1,j2 GrpΦj1,j2q be the

graph with vertices V j11 V j2

2 . The base case j2 0 is Proposition 2.24.

57

Let U Gj1,j2 be a largest maximal unlinked set, Ci tpu, iq P U | u P Gj1,j21u,and ci |Ci| for i P Y2, as above. Let p : Gj1,j2 ÝÑ Gj1,j21 be the projection dropping

the last coordinate.

We consider several cases:

Case 1: Suppose only one ci is nonzero. Note p pCiq is unlinked for any i. Hence

ci ¤ 3j12j21 3j12j2 by the inductive hypothesis.

Case 2: Suppose exactly two ci and cj are non-zero. Suppose i and j are linked. If

ci 3j12j21 then by the induction hypothesis p pCiq is a maximal unlinked set in Gj1,j21

of the form V W with V a type. This implies that for every u P Gj1,j21ppCiq the set

p pCiq contains both an element which is linked with u and an element which is unlinked

with u (see the construction in the proof of Lemma 2.25). But every element in p pCjqis linked with every element in p pCiq by Bj1,j21pppuq, ppvqq B0,1pi, jq 1, which is a

contradition. Hence ci 3j12j21. Thus |U | 3j12j2 .

Now suppose i and j are unlinked. Then |U | 3j12j2 if and only if p pCiq and p pCjqare maximal unlinked sets. But these must also be unlinked with each other and hence

p pCiq p pCjq by maximality. Then ppUq ppCiq V W for V a type and W a

maximal unlinked set. Thus U V pW ti, juq.Case 3: Suppose at least three of the ci are non-zero. Then for any unlinked pair

of indices ti, ju, there exists k such that ck 0 and at least one of ti, ku, tj, ku are

linked by the pigeonhole principle. Let pu, iq P Ci, pv, jq P Cj, and pw, kq P Ck. Then it

follows that WLOG Bj1,j21pu,wq B0,1pi, kq 1 and Bj1,j21pv,wq B0,1pj, kq 0.

So u v, which implies p is injective on Ci Y Cj. This implies that ci cj ¤ 3j12j21

with equality if and only if ppCi Y Cjq V W is an unlinked set of maximal size in

Gj1,j21 by the inductive hypothesis.

58

Then |U | pc0 c2q pc1 c3q ¤ 3j12j2 with equality if and only if ppC0 Y C2qand ppC1 Y C3q are unlinked of maximal size in Gj1,j21. But we also have that |U | pc0 c3q pc1 c2q ¤ 3j12j2 with equality if and only if ppC0 Y C3q and ppC1 Y C2q are

unlinked of maximal size in Gj1,j21. Suppose we have equality, and by the inductive

hypothesis suppose ppC0YC3q V0,3W0,3 and ppC0YC2q V0,2W0,2. Then ppC0q ppC0YC1qXppC0YC2q, so it follows that V0,3 and V0,2 must have the same type, as their

intersection is nontrivial and the types are mutually disjoint. In particular V0,3 V0,2.

By symmetry, we find that V0,3 V1,3 V1,2 V0,2. Since ppUq V0,3W0,3YV1,2W1,2

we find that ppUq V W for V V0,3 a type and W some unlinked set. But |W | 2j2 ,

and so it is a maximal unlinked set in V j22 .

The Constant

We have shown that an independent set of GrpΦj1,j2q has size at most 3j12j2 in Proposi-

tion 2.26. Theorem 1.6 then implies each partition with j1 k isOpXplogXq3j12j2j32j31q,so it follows that

¸d X,d1 mod 4

|SurjσpGalpKurKq, Hk8 q|

¸d X,d1 mod 4

|HomσpGalpKurKq, Hk8 q|

OXplogXq3k2ε

59

Theorem 1.6 then tells us that

¸d X,d1 mod 4

|HomσpGalpKurKq, Hk8 q|

1

2k

¸UγpUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

1

2k1

¸UγpUq

¸d X,d1 mod 4

3kωpdq

OXplogXq3k2ε

The only remaining step is to compute the leading constant, where

γpUq 23k¸

phuqPT pUqχk,0pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies

¹u`i

tu

$''&''%1 mod 4 i 0, 3

1 mod 4 i 1, 2, 4, 5

We will now prove

Proposition 2.27. ¸Uγ pUq 2k1.

Consider the vector space F3k

2 where each coordinate corresponds to an index in U .

Order the elements of U lexicographically, i.e. u pu1, ..., ukq for ui P t0, ..., 5u is ordered

by u ¤ v if there exists an i k such that for all j i uj vj and ui vi. We let each

y P F3k

2 correspond to a tuple of congruence classes phuq by

hu 1 mod 4 ðñ yu 1. (2.3)

Recursively define a matrix

60

Mk

~1 ~0 ~0

~0 ~1 ~0

~0 ~0 ~1

Mk1 Mk1 Mk1

where M1 I3 is the 3 3 identity matrix and ~0,~1 are row vectors of 0’s and 1’s

respectively.

For example for k 2 we get

M2

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

.

Then the y P F3k

2 satisfying condition p2.2.2q for an independent set U of type s P S are

solutions of

¸u:uij

yu

$''&''%1 if j 0, 3 and j P si

0 else

By construction

Mky ¸

u:uijyu

pi,jq

which makes these conditions equivalent to Mky w for an appropriate w P F3k2 . This

set of solutions is the coset y kerMk.

Now we will prove 2.27 by combining the following two lemmas.

61

Lemma 2.28. For all k ¥ 1 ¸Uγ pUq 23kkdim kerMk .

Proof. Recall

χk,0pphuquPV q k¹`1

1±

u`2,4 h2h4

,

and define λk : V Ñ F2 by λkpuq °k`1 λpu`q for λ the characteristic function of the

set t2, 4u. Then it follows that

χk,0pphuquPV q k¹`1

p1qλkpuqhu12 ,

Then

γpUq 23k¸phuq

¹u

p1qλkpuqhu12

¹tu,vu

p1qΦkpu,vqhu12

hv12

By the discussion above, for any phuq we let x P F3k

2 be the vector corresponding to it by

p2.3q. Then x belongs to a coset of kerMk, call it y kerMk. In particular, rephrasing

the congruence conditions p2.2.2q shows us that°

u:uia xu 1 if and only if a P t0, 3u.Now consider that Φpu, vq 0 if v P A t1, 3, 5u. U is a largest maximal unlinked

set, and so has a type s P S. It follows that

Φkpu,vq ¸i

Φpui, viq ¸

i:siBΦpui, viq

And similarly, λk|A 0 so that

λkpuq ¸i

λpuiq ¸

i:siBλpuiq

This way we show that

¸UγpUq 23k

¸sPS

¸xPykerMk

¹u

p1q°

i:siB λpuiqxu¹

tu,vup1q

°i:siB Φpui,viqxuxv

23k

¸sPS

¸xPykerMk

p1q°

u

°i:siB λpuiqxu

°tu,vu

°i:siB Φpui,viqxuxv .

62

Call UB Bk. Interchanging summations we can apply the binomial theorem to the

sum over types s P S to show

¸UγpUq 23k

¸xPykerMk

k¹j1

1 p1q

°uPUB

λpujqxu°tu,vuUB

Φpuj ,vjqxuxv.

Notice that for all j 1, ..., k we have

¸tu,vuUB

Φpuj, vjqxuxv ¸

uj0,vj2

xuxv ¸

uj0,vj4

xuxv ¸

uj2,vj4

xuxv

¸uj0

xu¸vj2

xv ¸uj0

xu¸vj4

xv ¸uj2

xu¸vj4

xv

and ¸uPUB

λpujqxu ¸uj2

xu ¸uj4

xu

and it follows that these are 0 from the conditions p2.2.2q. Thus

¸UγpUq 23k

¸xPykerMk

k¹j1

2

23kkdim kerMk .

Lemma 2.29. dim kerMk 3k 2k 1.

Proof. Without loss of generality suppose U has type s with s1 t0, 2, 4u. For x P F3k

2

and j P s1 let pj pxq be the projection onto F3k1

2 of the coordinates xu of x where u1 j.

Now x P kerMk if and only if°jPs1 pj pxq P kerMk1 and αk1ppj pxqq 0 for all

j P s1, with αk1 : F3k1

2 Ñ F2 the augmentation map defined by v ÞÑ 1 v.

It is clear that kerMk1 kerαk1. So we have that

kerMk #x |

¸jPs1

pj pxq P kerMk1, pj pxq P kerαk1

+.

63

There are | kerα3k1| choices for p0 pxq and p2 pxq. Then we have

p4 pxq P pp0 pxq p2 pxq kerMk1q X kerαk1.

That is p4 pxq belongs to a coset of kerMk1 kerαk1, so there are | kerMk1| choices.

So we have

| kerMk| | kerαk1|2| kerMk1| 22p3k11q| kerMk1|.

Clearly M1 is the identity map, and so kerM1 0. Then a simple induction shows that

| kerMk| 23k2k1 which completes the proof.

Combining Lemma 2.28 and Lemma 2.29 immediately proves Proposition 2.27. In

conclusion, plugging in the value of this constant yields the result from Theorem 2.23:

¸d X,d1 mod 4

|SurjσpGalpKurKq, Hk8 q|

1

4k

¸d X,d1 mod 4

3kωpdq OXplogXq3k2ε

We remind the reader that the proof of Theorem 2.23 for the other five families of

quadratic discriminants proceed similarly, and are addressed in more detail in the ap-

pendix.

2.2.3 A Corresponding Distribution

Rather than studying the asymptotic growth of the number of Hk8 -extensions, one could

attempt to prove that that the higher moments of |SurjσpGalpKurKq, H8q| give rise to a

distribution. Theorem 2.23 implies these moments are infinite, so if there is any hope of

getting a distribution we need to correct our function (similar to Gerth’s [27] corrections

to the original Cohen-Lenstra heuristics). We prove the following theorem:

64

Theorem 2.30 (A.-Klys [4]). Let pG1, Gq pH8 C2, H8q. For a quadratic field K with

discriminant d let f pdq be the number of everywhere unramified pG1, Gq-extensions of K

and let g pdq 3ωpdq. Then for all k P Z¥1 and pn,mq P tp4, 8q, p0, 8qu

E1,4

pfgqk

1

32

k

En,m

pfgqk

3

32

k

E1,4

pfgqk

1

192

k

En,m

pfgqk

1

64

k

where

En,mphq lim

XÑ8

°dn mod m,dPD

Xhpdq°

dn mod m,dPDX

1

Thus the function f pdq g pdq determines the point mass distribution on R in each family.

We mean by this last statement that the sequence of measures

µN,n,m pUq |tf pdq g pdq P U | d N, d n mod mu||td N | d n mod mu|

on R converges to the point mass µc in distribution for each pn,mq P tp1, 4q, p4, 8q, p0, 8qu,where c 132, 332, 1192, and 164 in the respective cases.

The proof is essentially the same as the proof of Theorem 2.23. We use the same

values for V,Φ, T,M, and χ as in Section 2.2.2, only now we use the value a 13. The

distribution converges to a point mass via the method of moments - point masses are

the unique distributions whose kth moment is ck for some c independent of k.

As a corollary we have the following result which ties back to the question of the

distribution of the Galois group of the maximal unramified extension.

65

Corollary 2.31. The density of quadratic fields K (ordered by absolute value of the

discriminant) with Gal pKurKq Hm8 is equal to 0 for any positive m P Z.

Proof. Theorem 2.30 gives

¸|d| X

Surjσ

Gur

Qp?dq, Hm8

kg pdqmk

¸|d| X

Surjσ

Gur

Qp?dq, H8

g pdq

mk o pXq .

Thus the distribution of the values ofSurjσ

Gur

Qp?dq, Hm8

g pdqm will again be a point

mass supported at some positive real number c by Theorem 2.23. By definition this

means that for any fixed m P Z and ε ¡ 0, one hundred percent of quadratic fields

satisfySurjσ

Gur

Qp?dq, Hm8

g pdqm c ε. In particular

Surjσ

Gur

Qp?dq, Hm8

is non-

zero one hundred percent of the time, which means there is at least one unamified

Hm8 -extension. But this holds for any m. The corollary follows.

2.3 Other Groups

The methods in Section 2.2 are far from unique. Lemmermeyer gave similar classifi-

cations for five other small 2-groups (see [40] and [41]). We will address one of these

groups, D4 the dihedral group of order 8, explicitly in Section 2.3.1. The other groups

will be encompassed in the work in Section 2.3.2, in which we use cohomological meth-

ods to generalize Lemmermeyer’s classifications to a much larger family of groups. The

author joint with Klys has a work in progress applying the Fouvry-Kluners sieve to this

classification in order to generalize the results of Section 2.2.2 to this larger family of

groups. We conclude this chapter in Section 2.3.4 by showing sufficient conditions for

Wood’s conjecture [59] to hold for admissible pairs of the form pG,G C2q, which are

in general weaker than previously known conditional results.

66

2.3.1 Dihedral Group of Order 8

Lemmermeyer gives a classification of unramified D4-extensions of quadratic fields anal-

ogous to that for H8:

Theorem ( [40]). Let k be a quadratic number field with discriminant d. There exists

an unramified extension Mk with GalpMkq D4, the dihedral group of order 8, which

is normal over Q if and only if

(a) GalpMQq D4 C2,

(b) there is a factorization d d1d2d3 into three discriminants (called a D4-factorization),

at most one of which is negative,

(c) for all primes pi | di we haved1

p2

d2

p3

1.

Moreover, for each such D4-factorization d d1d2d3 there are exactly 2ωpdq2 such ex-

tensions Mk.

The argument in Section 2.2.1 applies to (and in some ways is much easier to imple-

ment for) the following result:

Theorem 2.32 (A. [1]). EpD4q 8.

Proof. As in Section 2.2.1, define apd1, d2, d3q to be the number of such extensions

corresponding to the factorization d pd1qd2, d3 and

Dps, d1, d2q ¸

d1d2mdapd1, d2,mqds.

In the same vein as lemma 2.15 we find that, for some integer 0 δ ¤ 6 to account for

permutations,

Dps, d1, d2q pd1d2qs 1

8δ

¸a|d1

¸b|d2

d1

b

d2

a

¸m sqf

p1 χ4pmqqms

67

This series trivially has one simple pole at s 1 with residue

pd1d2q1 1

8δ

¸a|d1

¸b|d2

d1

b

d2

a

.

Consider the pairs d1, d2 with d1 a fixed odd prime 3 mod 4 and d2 d14x1 ¡ 0 for

any integer x which also makes d2 prime. Thus°a|d1

°b|d2

d1

b

d2

a

4 by quadratic

reciprocity. It then follows:

EpD4q ¥¸d1,d2

Ress1Dps, d1, d2q

¥¸

d2d14x1 prime

pd1d2q1 1

2δ

1

d12δ

¸d2d14x1 prime

pd2q1

8

by Dirichlet’s theorem on arithmetic progressions.

Following along the steps we took for the quaternion group, the next step would

be to count asymptotically how many unramified D4-extensions there are of quadratic

fields as in Theorem 2.23. Indeed the same methods will work, and we can apply

the Fouvry-Kluners sieve in Theorem 1.6. The set of vertices is t0, 1, 2, 3, 4, 5u and

the corresponding function Φ : t0, 1, 2, 3, 4, 5u2 Ñ F2 is the characteristic function of

tp2, 0q, p3, 0q, p0, 2q, p1, 2qu. This is a path on four vertices together with two isolated

points (the vertices 4and 5), so the largest size of an independent set is 4 and there are

three independent sets of that size.

Theorem 2.33.

NpD4, D4 C2;Xq cXplogXq3

68

The computation of the constant c is done similarly to Section 2.2.2, and can be

explicitly shown to be nonzero. We omit the proof of this result (and the corresponding

results for Dk4) for a few reasons. Firstly, it is qualitatively similar to the proof of

Theorem 2.23. Secondly, this result can be derived directly from the work of Fouvry-

Kluners on the 4-rank [25] due to the close relationship between the pairs pC4, D4q and

pD4, D4 C2q. Thirdly, the result is considered in recent work of Klys [39] expanding

the methods we used to prove Theorem 2.23 to a subfamily of pairs pG,G1q such that

G1 fits into an exact sequence

1 C2 G1 Cn2 1

We include this section to demonstrate that these methods are not unique to H8,

and give another motivating example for the work in Section 2.3.2 generalizing Lemmer-

meyer’s classifications.

2.3.2 Centrally Admissible Pairs pG,G1q

Lemmermeyer’s classifications for unramified H8- and D4-extensions were the key ingre-

dients in the proofs of number field counting Theorems 2.23 and 2.33. Lemmermeyer

gives similar classifications for three other small 2-groups in his dissertation [41], but

we can do much better. In this section we will provide classifications for unramified

pG,G1q-extensions for the following admissible pairs:

Definition 2.34. Call a pair pG,G1q centrally admissible if

• G G1

• rG1, G1s G

69

• G1 has nilpotency class 2, i.e. rG1, G1s ZpG1q

• G1 xy P G1 : xyy XG t1uy

If rG1 : Gs 2, then this pair is admissible in the sense of Wood’s definition [59].

Lemmermeyer’s classification of unramified H8-extensions of quadratic fields corresponds

to the pair pH8, H8 σ C2q as in Theorem 2.23. Likewise, Lemmermeyer’s classification

of unramified D4-extensions of quadratic fields corresponds to the pair pD4, D4C2q. In

this section we classify unramified pG,G1q-extensions, i.e. unramified G-extensions LKsuch that GalpLQq G1 and GalpKQq G1G. We prove the following result which

generalizes Lemmermeyer’s classifications:

Theorem 2.35 (A. [2]). Let pG,G1q be a central admissible pair, KQ an abelian

extension with f : GQ GalpKQq G1G, and g8 P pG1qab an element of or-

der dividing 2 with fpI8q xg8Gy. There exists an unramified pG,G1q-extension

LK with I8pLQqrG1, G1s xg8y ¤ pG1qab if and only if their exists a factorization

discpKqrG:rG1,G1ss ±yPY pdyqrpG

1qab:xyys into coprime discriminants (except that at most

two are allowed to be even) satisfying the following:

1. (a) Y ty P G1 : xyy XG 1urG1, G1s pG1qab

(b) g8 P Y

(c) |y| | pp 1qp8 for every prime p | dy.

(d) If dy, dy1 are even then WLOG of y has order dividing 2, 4 || dy, ry, y1sE 1,

and xy, y1y XGrG1, G1s 1. If dy is a unique even factor and 4 || dy, then y

has order dividing 2.

70

(e) The function f : GabQ ±

Ip Ñ pG1qab defined by

fpτpq

$'''''''''''&'''''''''''%

y p an odd prime with p | dy

y p 2 with 8 | dy

y p 1 with 4 || dy

y|y|2 or 1 p 1 and there exists a unique even factor 8 | dy

satisfies ¹p1 mod 4

fpτpq|fpτpq|2 g8

(f) p | dy implies yG fpτpq P G1G.

(g) pG1qab xy : dy 1y (i.e. f is surjective).

2. For every prime p | d¸yPY

p

dy

expprG1,G1sq

ry, fpτpqsG1 0

wherepdy

expprG1,G1sq

P Z expprG1, G1sqZ.

Moreover, for each such factorization there are exactly±yPY p#rG1, G1sr|y|sqωpdyq

#AutprG1sq#HompG1, rG1, G1sq

such extensions with rGalpLQqs rG1s P H2ppG1qab, rG1, G1sq, where rG1, G1srns tg PrG1, G1s : gn 0u is the n-torsion subgroup.

The proof of this theorem is addressed in two parts. We will first apply the solution

to the unramified embedding problem discussion in Section 1.2.1 to this scenario to

71

produce a “Lemmermeyer factorization” giving conditions on when a map f : GQ Ñ G

has an unramified lift rf : GQ Ñ E. Then, we apply the Lemmermeyer factorization

condition to centrally admissible pairs.

Lemmermeyer Factorization

In this section we will translate the solution to the embedding problem for central

extensions to conditions on discriminants in the specific case that G is abelian. Fix

an extension rEs P H2pG,Aq and the quotient map π : E Ñ G. Then r, sE : GGÑ A

is a well-defined bilinear map for all elements, not just the image of Frobenius and

inertia. For the remainder of this section, let unramified refer only to finite places.

Corollary 1.5 tells us that a continuous homomorphism f : GQ Ñ G has an unramified

lift rf : GQ Ñ E if and only if ResGfpIpqprEsq 0 and rfpFrobpq, fpIpqsE 0 for all primes

p (including the infinite prime).

For convenience, for p odd we will use τp for the generator of Iabp , and WLOG consider

f : GabQ Ñ G. For p 2 and 1, we let τ1 P I2 Z2Z Z2 be the generator

of I8pQpζ28qQq (i.e. the unique generator of the Z2Z factor) and τ2 a topological

generator of the Z2 factor, i.e. a topological generator of I2pQpζ28qQpiqq. We also use

discpfq to denote the discriminant of K Qker f.

Lemma 2.36. Let YE tg P G : ResGxgyprEsq 0u and g8 P YE of order dividing

2. Then there is a one-to-one correspondence between maps f : GabQ Ñ G satisfying

ResGfpIpqprEsq 0 and fpI8q xg8y with discpfq d ±pep and factorizations

discpfq ±yPYEpdyqrfpGQq:xyys into coprime discriminants (except for at most two allowed

to be even) where

72

1. |y| | pp 1qp8 for every prime dividing dy where |y| is the order of y,

2. If dy, dy1 are even then WLOG y has order dividing 2, 4 || dy, and ry, y1sE 1. If

dy is a unique even factor and 4 || dy, then y has order dividing 2.

3.

¹yPY

¹p|dy ,p1 mod 4

y|y|2

$''''''&''''''%

g8 2 - dy for all y P Y

y|y|2g8 4 || dy

y|y|2g8 or g8 there is a unique even factor 8 | dy

given by

fpτpq

$'''''''''''&'''''''''''%

y p an odd prime with p | dy

y p 2 with 8 | dy

y p 1 with 4 || dy

y|y|2 or 1 p 1 and there exists a unique even factor 8 | dy

respectively.

This lemma is a consequence of Kronecker-Weber. The proof is somewhat long and

cumbersome, so we will first demonstrate the content of this theorem with an example

from genus theory both to motivate the result and convince any reader not interested in

trudging through the proof.

Example 2.37. Since

GabQ 2Gab

Q xτ2,0, τ2,1, τp p an odd prime : τ 22,0 τ 2

2,1 τ 2p 1y

73

a Cn2 -extension of odd discriminant d corresponds to a homomorphism

f : xτp, p | d : τ 2p 1y Ñ Cn

2 .

We can produce a factorization from this map as follows: let dy ±

p:fpτpqy p where

p p1q p12 and d0 1. Then d ±

yPCn2d|im f |2y because the exponent of every

prime dividing d must be |im f | divided by the ramification degree, which is 2. The field

Qp?a,?bq for a, b P Z odd and squarefree corresponds to the factorization d pabq2 a2 b2 dv1 dv2 for v1, v2 a basis of C2

2 . This gives a one-to-one correspondance between

maps f : GQ Ñ Cn2 of discriminant and factorizations d ±

yPCn2d|im f |2y , since any

such factoriztion defines a map by sending τp ÞÑ y if p | dy. In this case, conditions 1

and 2 on the factorizations are trivially satisfied because |y| 2 and d is odd.

For this lemma, we are restricting the possible images of fpτpq to a specific sub-

set of elements YE G and giving a correspondence between these maps and certain

factorizations of the discriminant. This subset is specifically chosen so that we always

have ResGfpIpqprEsq 0, one of the conditions that was necessary to solve the embedding

problem. Condition 3 is then equivalent to forcing the generator τ8 P I8 to satisfy

fpτ8q P YE.

Proof of 2.36. Define dy : ±fpτpqyppqap a discriminant where paprfpGQq:xyys || discpfq

exactly divides discpfq for y 0, and d0 1. These are all clearly coprime. There exists

a continuous homomorphism fy : GabQ ±

Ip Ñ xyy which equals f on Ip for primes

with fpτpq y and 0 on all other primes. Let Ky Qker fy. Then discpKyq dy by

the higher ramification formula for discriminants and rKy : Qs |y|. It follows that±KyK is an unramified extension, where the power of p dividing discp±Kyq is exactly

74

apr±Ky : Kys app

± |y1|q|y| ap±

y1y |y1|. Thus we have¹yPYE

pdyq±

y1y |y1| discp¹

Kyq

discpKqr±Ky :Ks

discpfq1

|fpGQq|± |y|

Raising both sides to the power|fpGQq|± |y| yields the result up to a sign. The conditions

on dy and y follow immediately from fpτ1q having order dividing 2, fpτ1q commuting

with fpτ2q, and |fpτpq| | pp 1qp8.

Given such a factorization, define fpτpq by the following rules:

fpτpq

$'''''''''''&'''''''''''%

y p an odd prime with p | dy

y p 2 with 8 | dy

y p 1 with 4 || dy

y|y|2 or 1 p 1 and there exists a unique even factor 8 | dyThe relation to g8 in part 3 forces only one choice for fpτ1q to be acceptable, even in

the last case. This uniquely defines a map f : GabQ ±

Ip Ñ G given by the generators

τp. It is well-defined by conditions on the order of y, and conditions on even factors.

Moreover, ResGfpIpqprEsq 0 by fpτpq P YE for all finite primes. As in the other direction,

the field±Ky gives the factorization up to a sign.

We have now proven the theorem up to a sign, except for part 3 and the infinite

place. We will first show that part 3 is equivalent to ResGfpI8qprEsq 0, then show that

signs of the factorization also match.

Given such a factorization, the signs are not an obstruction to constructing a map

as GabQ ±

Ip a product over finite primes and we can get a map just coming from the

75

finite primes present in the factorization. ResGfpI8qprEsq 0 if and only if

fpτ1q¹

p3 mod 4

fpτpq|fpτpq|2 P YE

This is the same thing as g8 P YE, which is true by assumption.

As for the signs of discriminants, sgnpdiscpfqq p1qr2 for r2 the number of complex

places of Kker f . It similarly follows that sgnpdyq p1qr2pyq for r2pyq the number of

complex places of Ky. Therefore it suffices to show

r2pKq ¸yPYE

r2pyqrfpGQq : xyys mod 2

Depending on the reamification of 8 in K or Ky, we get the following values for r2

r2pKq

$''&''%0 8 not ramified

12rK : Qs else

r2pyq

$''&''%0 8 not ramified in Ky

12|y| else

We will break this into a few cases. Suppose first that the Sylow 2-subgroup of fpGQqis not cyclic. Then 2 | rfpGQq : xyys for all y P YE and 4 | |fpGQq| rK : Qs. Therefore

¸yPYE

r2pyqrfpGQq : xyys 0 r2pKq mod 2

Now suppose the Sylow 2-subgroup of fpGQq is cyclic, and generated by x P G. If y P YEis not a generator of xxy, then 2 | rfpGQq : xyys. Therefore it suffices to show that

r2pKq ¸

xzyxxyr2pzq

WLOG, choose the generators τp P IppQabQq so that if xfpτpqy xxy, then fpτpq x.

This implies that if xzy xxy and Kz Q, then z x. Therefore it suffices to show

76

that r2pKq r2pxq mod 2, by r2pQq 0. 8 is ramified if and only if it is ramified in

the Sylow 2-subgroup xxy because |I8| 2. The Sylow 2-subgroup has a splitting map,

which yields Kx ¤ K as a subfield of odd index. Therefore KKx must be unramified

over 8, and r2pKq rK : Kxsr2pxq r2pxq mod 2.

The field±

yPYE Ky with Galois group±

yPYExyy over Q is going to be very important.

Call this field L with Galois group H over Q. The following lemma follows immediately

from properties of the quotient map H Ñ G sending y ÞÑ y for each y P YE:

Lemma 2.38. r, sE : G G Ñ A lifts to a map r, sE : H H Ñ A by applying

r, sE coordinatewise and adding up the results, i.e. satisfies the following commutative

diagram

H H

GG A

r,sE

r,sE

In particular, if g : GQ ѱ

yPYExyy is defined by τp ÞÑ fpτpq P YE, then rfpFrobpq, fpτpqsE rgpFrobpq, gpτpqsE.

Thus it suffices to check the commutator condition rfpFrobpq, fpIpqsE 0 in±

yPYE Ky.

Lemma 2.39. Let f : GQ Ñ G be the continuous homomorphism of discriminant

discpfq ±pep corresponding to a factorization discpfq ±

yPYEpdyqrim f :xyys and let

g : GabQ Ñ H be the map sending τp ÞÑ fpτpq P YE. Then

rgpFrobpq, gpτpqsE ¸yPYE

¸q|dy

p

qbq

exppAq

ry, gpτpqsE

where

p

qbq

nP ZnZ is the image of p under the quotient map

Zq Ñ Zq pZq qpn,qbq1pq1qq

ãÑ ZnZ,

bq eqrfpGQq : xyys, and exppAq is the exponent of the group A.

77

Proof. We have that the Frobenius element Frobp P GabQ ±

Iq is ppqq the equivalence

class of p in each coordinate. Therefore, under g we get that

gpFrobpq g

¸q|discpfq

ppq.τq

¸yPYE

g

¸q|dyppq.τq

¸yPYE

¸q|dyppq.y

The equivalence class of p, denoted above by ppq acts on xyy by multiplication as an

element of Z|y|Z, namely

p

qeqrfpGQq:xyys

|y|

for each q | dy. The proof then follows from

the bilinearity of r, sE.

Definition 2.40. We defineba

n

: Z Ñ ZnZ byba

n °

qeq ||a

qeq

n

for any pa, bq 1 where a has prime factorization

±qeq .

We can now prove the main theorem for generalized Lemmermeyer factorizations as

an immediate consequence of the work in this section:

Theorem 2.41. Let YE tg P G : ResGxgyprEsq 0u and g8 P YE. Then the homomor-

phism f : GabQ Ñ G with fpI8q xg8y lifts to rf : GQ Ñ E unramified at finite places

if and only if there exists a factorization discpfq ±yPYEpdyqrfpGQq:xyys into coprime

discriminants (except for at most two allowed to be even) where

1. |y| | pp 1qp8 for every prime dividing dy where |y| is the order of y,

2. If dy, dy1 are even then WLOG y has order dividing 2, 4 || dy, and ry, y1sE 1. If

dy is a unique even factor and 4 || dy, then y has order dividing 2.

78

3. The function f : GabQ ±

Ip Ñ pG1qab is related to the factorization by

fpτpq

$'''''''''''&'''''''''''%

y p an odd prime with p | dy

y p 2 with 8 | dy

y p 1 with 4 || dy

y|y|2 or 1 p 1 and there exists a unique even factor 8 | dy

and satisfies ¹p1 mod 4

fpτpq|fpτpq|2 g8

4. For every prime p | d

¸yPY

p

dy

expprG1,G1sq

ry, fpτpqsG1 0

wherepdy

expprG1,G1sq

P Z expprG1, G1sqZ.

Proof. Parts 1, 2, and 3 are equivalent to ResGfpIpqprEsq 0 by Lemma 2.36.

For part 2, we have that q | dy1 if and only if y1 gpτqq. Then

¸yPYE

p

dy

exppAq

ry, fpτpqsE ¸yPYE

¸q|dy

p

qbq

exppAq

ry, gpτpqsE

rgpFrobpq, gpIpqsE

rfpFrobpq, fpIpqsE

Which is 0 if and only if rfpFrobpq, fpIpqsE 0.

This is equivalent to the existence of a lift rf unramified at all finite places by the

solution to the embedding problem discussed in Corllary 1.5.

79

Proof of the Classification

In this section, we translate the Lemmermeyer factorizations in Theorem 2.41 to prove

the classification of pG,G1q-extensions of Q given in Theorem 2.35.

Proposition 2.42. Let pG,G1q be a central admissible pair. Then AutprG1, G1sq acts

transitively on trEs P H2ppG1qab, rG1, G1sq : E G1u with stabilizers isomorphic to

AutprG1sq

Proof. Given rEs with an embedding i : rG1, G1s ãÑ E, α P AutprG1, G1sq acts on it by

i ÞÑ iα. Because rG1, G1s rE,Es is the commutator subgroup, every such isomorphism

rEs rEs must arise from this action, showing that the stabilizer is a quotient of

AutprEsq. Moreover, every extension is completely determined by the embedding of

rG1, G1s as the commutator subgroup, because the quotient is just the abelianization

map which is characteristic (i.e. invariant under any automorphism which is trivial

on the abelianization). Therefore the the stabilizer is isomorphic to AutprG1sq and

because AutprG1, G1sq acts transitively on IsomprG1, G1s, rE,Esq, it must act transitively

on trEs : E G1u.

Proof of Theorem 2.35. Fix a choice of rG1s P H2ppG1qab, rG1, G1sq. We will prove the if

and only if conditions first.

Because pG,G1q is central and admissible, we know that G1 is a central rG1, G1s-extension of pG1qab. Setting A rG1, G1s, E G1 gives us the set-up of the Theorem 2.41.

The map f and Theorem 2.41 imply the existence of a surjective map f : GQ Ñ pG1qab

such that M Qker fis unramified over K with fpI8q xg8y (in the case that there are

two even factors, 2 is unramified in MK because xy, y1y XGrG1, G1s 1 for dy, dy1 the

two even factors). Theorem 2.41 also implies that an unramified extension rf : GQ Ñ E

80

exists if and only if there is a factorization discpMQq ±yPY pdyqrpG

1qab:xyys satisfying

parts 1 and 2.

It then suffices to check whether or not rf is surjective by Galois theory. We have f

is surjective by construction, so rfpGQq ¤ G1 is a subgroup which surjects onto pG1qab.Therefore G1 rfpGQqA as A rG1, G1s. If A ¤ rfpGQq then we are done. Let B AX rfpGQq. Then we have G1B rfpGQqBAB since rfpGQqBXAB 1. However,rfpGQqB rfpGQqp rfpGQq XAq rfpGQqAA G1A is abelian, implying that G1B is

abelian. Thus rG1, G1s A ¤ B rfpGQq X A, concluding the proof for fixed rG1s.Because rG1, G1s is the commutator subgroup, Proposition 2.42 tells us that different

choices of rG1s only differ by embedding rG1, G1s up to automorphism. This implies that

r, sG1 differs up to automorphism and RespG1qabxyy prG1sq is fixed up to isomorphism. This

implies that Y is independent of the choice of rG1s and r, sG1 is fixed up to automorphism

of rG1, G1s, for which the equation in part 2 is invariant because automorphisms of rG1, G1sare Z expprG1, G1sqZ-linear maps. Therefore parts 1 and 2 are invariant of the choice of

extension class rG1s P H2ppG1qab, rG1, G1sq.Any unramified pG,G1q-extension LK corresponding to the factorization

discpKqrG:rG1,G1ss ¹yPYG1

pdyqrpG1qab:xyys

contains the field rK given by the continuous homomorphism f : GabQ Ñ pG1qab with

fpτpq y for p | dy by construction, and fpI8q xg8y by assumption. So it suffices to

count unramified (at finite places) rG1, G1s-extensions of rK whose corresponding exten-

sion isomorphism class is rG1s. By assumption rG1, G1s is abelian and central, so any such

extension must come from a surjective homomorphism Clp rKq Ñ rG1, G1s. Let Hp rKq be

81

the Hilbert class field of rK. Then the exact sequence

1 Clp rKq GalpHp rKqQq Galp rKQq 1

gives rise to the following inflation-restriction sequence with trivial action on rG1, G1s:

0 H1pGalp rKQq, rG1, G1sq H1pGalpHp rKqQq, rG1, G1sq

H1pClp rKq, rG1, G1sqGalp rKQq H2pGalp rKQq, rG1, G1sq

where the last map sends φ : Clp rKq Ñ rG1, G1s to rGalpHp rKqkerφQqs. By assumption

rG1s is in the image of this map. So it follows that there are

|H1pGalpHp rKqQq, rG1, G1sq||H1pGalp rKQq, rG1, G1sq|

such homomorphisms Clp rKq Ñ rG1, G1s giving rise to rG1s. Notice that because rG1, G1s is

abelian, any homomorphism GalpHp rKqQq Ñ rG1, G1smust factor through Galp rKgenQq,the Galois group of the genus field. If fp : Gab

Q Ñ xyy is the map sending τp ÞÑ y and

τq ÞÑ 0 for any prime q p then we can describe the genus field as±

p|discpKqQker fp

with

Galois group±

p|discpKqxfpτpqy over Q. Thus

H1pGalpHp rKqQq, rG1, G1sq Hom

¹p|discpKq

xfpτpqy, rG1, G1s

¹

p|discpKqHompxfpτpqy, rG1, G1sq

¹

p|discpKqrG1, G1sr|fpτpq|s

¹yPY

¹p|dyrG1, G1sr|y|s

82

where Grns denotes the n-torsion of an abelian group. (Note: if fpτ1q y, then we

must also count 1 in ωpdyq). So it follows that

#H1pGalpHp rKqQ, rG1, G1sq ¹yPYp#rG1, G1sr|y|sqωpdyq

Then the number of such homomorphisms Clp rKq Ñ rG1, G1s giving rise to rG1s is±yPY #rG1, G1sr|y|sωpdyq

#H1pGalp rKQq, rG1, G1sq

±yPY #rG1, G1sr|y|sωpdyq

#HomppG1qab, rG1, G1sqSince every homomorphism G1 Ñ rG1, G1s must factor through pG1qab, we can replace

the bottom with HompG1, rG1, G1sq.Normally, there are more homomorphisms than corresponding fields, so one might

expect that we need to divide by AutprG1, G1sq to eliminate redundancy. However, the

redundancy coming from automorphisms of rG1, G1s in this case can produce different

isomorphism classes of rG1s. Proposition 2.42 shows that the subgroup AutprG1sq ãÑAutprG1, G1sq parametrizes all the redundancy, concluding the proof.

2.3.3 Example: Heisenberg Groups

Fix an odd prime ` and consider the Heisenberg group

Hp`3q xx, y : rx, ys z, x` y` z` 1, rx, zs ry, zs 1y

This is the unique nonabelian group of order `3 and exponent `. We define E Hp`3qC`by the following presentation:

xx, y, σ : rx, ys rx, σs ry, σs z, x` y` z` σ` 1, rx, zs ry, zs 1y

In particular, xzy rE,Es ¤ ZpEq giving us a central exact sequence

1 C` E C3` 1

83

This makes pHp`3q, Eq a central admissible pair. Moreover, because every element of E

has order `, it follows that Y Eab C3` . We will prove the following classification as

an example of Theorem 2.35:

Theorem 2.43. Let KQ be a cyclic degree ` extension of discriminant d p`1q`1r`1

coprime to `. Fix a generator g P GalpKQq. Then there exists an unramified pHp`3q, Eqextension LK if and only if there exists A,B,C P Z`Z with AB C 0 and

p

q`1

`

pAq p

r`1

`B 0

q

p`1

`

A q

r`1

`pCq 0

r

p`1

`

pBq

r

q`1

`

C 0

Moreover, in that case there are exactly ` 1 such extensions for each factorization,

which all have distinct extension classes rGalpLQqs P H2pC3` , C`q.

Proof. Fix π : Eab Ñ C` the quotient map given above. Without loss of generality,

we choose generators τp, τq, τr P GabQ such that f : Gab

Q Ñ GalpKQq C` satisfies

fpτpq fpτqq fpτrq g.

We will walk through the conditions of the Theorem 2.35 for this case, describing

what they each correspond to. Suppose we have a factorization

discpKqrHp`3q:rE,Ess d`2

¹yPYpdyqrEab,xyys

¹yPYE

d`2

y

in other words, d ±yPYE dy. Then:

1. (a) Y is the set of possible images of ramification in Eab, which is equivalent to

having ResEab

fpIpqprEsq 0 for finite primes. We see that every element of E

has order `, which implies that YE Eab C3` .

84

(b) g8 P Eab has order dividing 2 in an `-group, so g8 1 P Y .

(c) By the existence of the field K, we must have ` | p 1, q 1, r 1. Every

element of Y has order `, so this part is vacuous.

(d) 2 cannot be ramified in K, so this part is vacuous.

(e) All elements have odd order, so fpτpq|fpτpq|2 1 for all p and g8 1, making

this part vacuous. This part shows that ResEab

fpI8qprEsq 0.

(f) By assumption, fpτpq fpτqq fpτrq g, so dy 1 requires πpyq g. This

is necessary for there to exist f : GQ Ñ Eab a well-defined lift of f : GQ Ñ G

corresponding to our factorization. Call the corresponding y values fpτpq yp, fpτqq yq, and fpτrq yr respectively.

(g) This is equivalent to the lift f : GQ Ñ Eab being surjective, i.e. Eab xyp, yq, yry. This is true if and only if kerπ xypy1

q , ypy1r y, i.e. ypy

1q , ypy

1r

are linearly independent by dimF`kerπ 2.

This implies that part 1 of Theorem 2.35 is satisfied if and only if the factorization is

given by d ±πpyqg dy and typy1

q , ypy1r u are linearly independent. Given these, part

1 gives us a lift f : GQ Eab with ResEab

fpIpqprEsq 0 for all primes.

2. This is equivalent to rfpFrobpq, fpIpqsE 0, and can be distilled into three equa-

tions: p

q`1

`

ryq, ypsE p

r`1

`ryr, ypsE 0

q

p`1

`

ryp, yqsE q

r`1

`ryr, yqsE 0

r

p`1

`

ryp, yrsE

r

q`1

`

ryq, yrsE 0

85

since the factors will be trivial whenever dy 1.

Theorem 2.35 tells us that there exists an unramified pHp`3q, Eq-extension LK if

and only if parts 1 and 2 hold.

Consider the presentation given above

E xx, y, σ : rx, ys rx, σs ry, σs z, x` y` z` σ` 1, rx, zs ry, zs 1y

and without loss of generality let σ be a preimage of g. Then we have the bilinear map

r, sE : xx, y, σ : x` y` σ` 1yab Ñ xzy Z`Z

where we write Z`Z additively. r, sE is defined by its values on the basis x, y, σ, given

by being antisymmetric with rx, ysE rx, σsE ry, σsE z.

By construction, we must have yp vpσ, yq vqσ, and yr vrσ for vi P xx, yy. We

have that ryi, yjsE rvi, vjsE rvi, σsE rσ, vjsE for any choice of i, j P tp, q, ru by a

simple computation. Consider

rxa1yb1 , xa2yb2sE a1b2 a2b1 det

a1 b1

a2 b2

rxa1yb1 , σsE a1 b1

So it follows that

rxa1yb1σ, xa2yb2σsE a1b2 a2b1 a1 b1 a2 b2

pa1 1qpb2 1q pa2 1qpb1 1q

is also a determinant. The requirement that ypy1q , ypy

1r are linearly independent is the

same as the requirement that vpv1q , vpv

1r are linearly independent. A pair of vectors

86

being linearly independent is equivalent to the matrix with those vectors as column

vectors having nonzero determinant. So condition 1 is equivalent to yp, yq, yr mapping

to g under π and satisfying

0 rvpv1q , vpv

1r sE

rvp, vrsE rvq, vpsE rvq, vrsE

rvr, vpsE rvp, vqsE rvq, vrsE

ryr, ypsE ryp, yqsE ryq, yrsE

Fix A,B,C P Z`Z with A B C 0. Suppose yp, yq are chosen with ryp, yqsE A.

Then we have that choosing yr xarybrσ with ryr, ypsE B and ryq, yrsE C is

equivalent to solving a system of two linear equations in two variables. Write yp xapybpσ and yq xaqybqσ, then we want a solution to

par 1qpbp 1q pap 1qpbr 1q B

paq 1qpbr 1q par 1qpbq 1q C

In other words bp 1 pap 1qpbq 1q aq 1

arbr

B bp ap

C aq bq

where the matrix has determinant

pbp 1qpaq 1q pap 1qpbq 1q ryq, ypsE A

WLOG by symmetry we may assume that A 0 by A B C 0, therefore we may

find such a choice for yr. This shows that any such choice of A,B,C can be realized

87

by yp, yq, yr, so the equations on Dirichlet characters in this theorem are equivalent to

condition 2 of Theorem 2.35 for some choice of yp, yq, yr satisfying condition 1.

As for the number of such extensions corresponding to a factorization, notice that

plugging rE,Es C`, |y| `, and Eab C3` into the formula in Theorem 2.35 tells us

that there are

`ωpdq3

|AutprEsq| 1

|AutprEsq|such extensions for the choice of rEs. This number must be a nonnegative integer

implying that AutprEsq 1, which can be checked group theoretically. There are

#AutprE,Esq#AutprEsq p` 1q1 isomorphism classes of rEs by Proposition 2.42,

concluding the proof.

2.3.4 Admissible Pairs of the Form pG,G C2q

There is one case in which we can say something about EpG,G1q in greater generality,

and that is when the group G is generated by elements of order 2. In this case G has

a trivial admissible pair pG,G C2q. Any unramified extension of a quadratic field

corresponding to this pair is then a compositum of the quadratic field and some field

K with Galois group G over Q whose inertia groups are all cyclic of order 1 or 2. Call

any field KQ whose inertia groups are cyclic of order 1 or 2 quadratically ramified.

How much room does this extra freedom give us?

Lemma 2.44. Suppose KQ is a Galois extension with Galois group G such that

Kp?dqQp?dq is unramified for d a quadratic discriminant. Then

#tKp?aq : unramified over Qp?aq and a X a quadratic discriminantu

88

is bounded below by

p1 op1qq 6

|d|π2X

as X Ñ 8.

Proof. For any quadratic discriminant a coprime to d (with Qp?adq ¤ K) and L Kp?dq we have the following field diagram:

Q

Qp?dq K

L

Qp?adq

Qp?a,?dq

Lp?aq

Kp?adq

C2

G

G

C2

C2

C2

C2

C2

G

C2 G

Every field in this diagram is clearly Galois over Q, and for each pair of fields in

this diagram their meet and join in the diagram are given by their intersection and

compositum respectively.

By assumption LQp?dq is unramified, which implies Lp?aqQp?a,?dq is unrami-

fied. Genus theory implies Qp?a,?dqQp?adq is unramified. Putting those two together

shows that Lp?aqQp?adq is unramified. In particular, Kp?adqQp?adq is an unrami-

fied subsextension with Galois group G.

89

As a consequence, given such a K and odd d, this construction gives asymptotically

¸|a| X:pa,dq1

1 Ress1

¸a:pa,dq1

|a|s X

unramified extensions M of quadratic fields with Galois group G and Galois group

G C2 over Q such that MC2 K, by summing over quadratic discriminants a X

and applying a Tauberian theorem. By manipulating the Euler product of ζpsq we find

that

¸a:pa,dq1

|a|s

$''&''%p1p4qs2p8qsq

p12sq±p|dp1psqζpsqζp2sq d odd

1±p|dp1psq

ζpsqζp2sq d even

The residue at s 1 is±

p|dp1 p1q1 6π2 in both cases. We have that for d odd

¹p|dp1 p1q1 6

π2 |d|±

p|dpp 1q6

π2

¥ |d|±p|d p

2

6

π2

6

|d|π2

For even d the work is similar, noting that±

p|d p d.

It then follows that

#tKp?aq : unramified over Qp?aq for a X a quadratic discriminantu6

|d|π2X

is bounded below by ¸|a| X:pa,dq1

1

6|d|π2X

¥ 1 op1q

as X Ñ 8.

90

Counting unramified extensions of quadratic fields with Galois group G becomes a

question of counting pairs of pK, dq with d minimal such that Kp?dqQp?dq is unrami-

fied.

Lemma 2.45. Suppose KQ is quadratically ramified with Galois group G and discrim-

inant DK. Let dp be the discriminant of a quadratic field kQ unramified at all finite

primes q p such that the corresponding local field kp ¤ Kp over Qp if p | DK and dp 1

otherwise. Then d ±dp is a quadratic discriminant, and we have Kp

adqQp

adq is

unramified.

Proof. Clearly d is squarefree up to a 4 or 8, so it suffices to check the sign in the odd

case to show it is a quadratic discriminant. But each dp 1 mod 4 for p odd, and

therefore so is their product. We should note that any two totally ramified quadratic

subextensions of Kp for p odd must have the same discriminant, which is easily seen by

examining the fact that they all share a common unramified extension.

It suffices to show the inertia subgroups at primes dividing d ofGC2 GalpKQpadqQq

are all of order 2, because they are all nontrivial in the C2-quotient. Notice how

Qpadq b Qp has discriminant equal to dp, which follows from examining the polyno-

mial x2 d over Qp. From facts about extensions of Qp, there are exactly 2 quadratic

extensions having discriminant dp, at least one of which is a subextension of Kp. If

Qpadq b Qp is a totally ramified quadratic subextension of Kp, then we are done as

Kpadq b Qp Kp which is quadratically ramified. Otherwise there is another totally

91

ramified quadratic extension L ¤ Kp. Consider the following diagram:

Qp

QppadqL

Kp Lpadq

Kppadq

22

urur

ur

where Lpadq being unramified over each quadratic subfield follows from the fact that,

given the unique unramified quadratic extension M of Qp, Lpadq ML Mp

adq.

But this implies the remaining unlabeled extensions must also be unramified, showing

that #Ip 2.

Thus for a quadratically ramified extension KQ, the minimal d for which Kp?dqis unramified over Qp?dq is the one ramified at exactly the same finite primes as K. So

it follows that counting unramified G-extensions of quadratic fields with Galois group

GC2 over Q is equivalent to counting quadratically ramified extensions K with Galois

group G ordered by dK .

This falls immediately into the area of number field counting problems similar to

Malle’s conjecture [42] [43], which were discussed in Section 1.1.2. Similar asymptotics

are expected to be true if we count G-extensions KQ with certain kinds of restricted

ramification (see Ellenberg-Venkatesh [24] and Wood [59] for examples). KQ being

92

quadratically ramified is a restriction on ramification, so we can ask if Malle’s conjec-

ture is true for counting G-extensions KQ which are quadratically ramified and have

discriminant less than X.

Taking the asymptotic from Malle’s conjecture and combining it with the asymptotics

from lemmas 2.44 and 2.45 we would expect that EpG,G C2q 8. We will prove a

stronger statement, which assumes something strictly weaker than Malle’s conjecture:

Corollary 2.46. Let G be a group generated by elements of order 2. Suppose the number

of quadratically ramified G-extensions KQ ramified with discriminant bounded by X is

¥ CX2#GplogXq1 for sufficiently large X and some positive constant C ¡ 0. Then

EpG,G C2q 8.

We remark that Malle’s conjecture predicts an asymptotic of CX2#GplogXqb for

some b ¥ 0, stronger than the bound assumed above.

Proof. The discriminant in the usual sense, dK , is equal to the discriminant d given by

the regular representation G ãÑ Sn for n #G. Then for any quadratic discriminant

a such that Kp?aqQp?aq is unramified, Kp?aq has discriminant a#G. So it follows

that |dK | |dK |#G2. This implies that there are ¥ CX logpXq1 quadratically ramified

G-extensions KQ with dK X for sufficiently large X.

For any quadratically ramifed KQ there are ¥ p1 op1qq 6

|dK |π2X quadratic fields k

with discriminant dk X such that Kkk is unramified. The expected number then

satisfies the following bound:

EpG,G C2q ¥ 6

π2limXÑ8

¸K quad. ram., dK X

1

|dK |

93

Although we do not do so here, keeping track of the sign of the discriminant only

slightly changes the values 6π2. To conclude the proof we only need to show that this

sum diverges.

Consider the following identity (inspired by the relationship between Mellin trans-

forms and Dirichlet series):» N

1

¸|dK | x

1

dx

x2

¸|dK | N

» N

|dK |

dx

x2

¸

|dK | N

1

x

N|d|K

¸

|dK | N

1

|dK | 1

N

¸|dK | N

1

In this computation, each summation is finite allowing us to switch the order of integrals

and summations. Therefore we can express the summation we are interested in studying

as an improper integral

limNÑ8

¸|dK | N

1

|dK |» 8

1

¸|dK | x

1

dx

x2 lim

NÑ81

N

¸|dK | N

1

¥» 8

1

¸|dK | x

1

dx

x2

Let N0 ¥ 1 be a number such that #t|dK | Xu ¥ XplogXq1 for all X ¥ N0. Applying

this to the improper integral shows that

limNÑ8

¸|dK | N

1

|dK |¥» N0

1

¸|dK | x

1

dx

x2» 8

N0

¸|dK | x

1

dx

x2

¥» 8

N0

¸|dK | x

1

dx

x2

¥» 8

N0

dx

Cx log x

94

It then suffices to show that this integral diverges, which follows from a simple calculus

exercise: » 8

N0

dx

Cxplog xq » 8

logN0

du

Cu

C1 rlog us8logN0

C1 limBÑ8

logpBq C1 log logN0

8

Remark: One could also just use the heuristic in Ellenberg-Venkatesh [24] for Malle’s

conjecture with restricted ramification to count G C2-extensions, which predicts that

the number of G C2-extensions with discriminant X unramified over its quadratic

subfield is asymptotic to cX logpXqbpQ,GC2q for sufficiantly large X and some positive

constant c. From this point, knowing bpQ, GC2q ¡ 0 is enough to conclude an infinite

expected number. It is known, however, that in certain cases the value for bpQ, Gq in

Malle’s Conjecture and related heuristcs is incorrect (a counterexample is proven by

Kluners [34]). The benefit of the above proof is that it is independent of the actual

value for bpQ, Gq, and assumes a far weaker asymptotic in general. Bhargava proves

EpSn, Sn C2q 8 by proving Malle’s conjecture for Sn, n ¤ 5 and then using the

above method [6].

What cases do we know this result holds? Kluners and Malle [37] proved that for G

in the regular representation if there exists a G-extension KQ, then

#tKQ : DKQ Xu ¥ cX`

#Gp`1q ,

95

where ` is the smallest prime dividing the order of the center, #ZpGq. Their proof

immediately implies that if 2 | #ZpGq and there exists a quadratically ramified G-

extension KQ then

#tKQ : quadratically ramified DKQ Xu ¥ cX2#G

Corollary 2.47. For any 2-group G, EpG,G C2q 8.

Proof. Kluners-Malle’s result and Corollary 2.46 together imply that it suffices to show

that there exists a quadratically ramified G-extension KQ. G is a 2-group, so it has a

lower central series. We can then build up from a quotient of G isomorphic to C2 all the

way to G by using the embedding problem for central extensions as in Serre’s Topics in

Galois Theory [53], and so long as sufficiently many primes are ramified we can ensure

that it is quadratically ramified at every step.

96

Chapter 3

Number Field Counting

Let K be a number field, GK GalpKKq the absolute Galois group, P the set of

places of K, and IK the set of ideals of the ring of integers in K. For each p P P let

Dp GalpKpKpq be the absolute decomposition group and Ip Dp the absolute inertia

group. These are defined as subgroups of GK up to conjugacy.

We will consider number field counting problems in any sufficiently nice ordering.

We take some cues from orders we want to be consider, such as the discriminant or

conductor, and define admissible invariants to depend only on the ramification data:

Definition 3.1. Call a function inv :±

p HompDp, Gq Ñ IK admissible if the following

hold for γ pγpq:

1. p | invpγq if and only if γppIpq 1.

2. γp|Ip γ1p|Ip for all Archimedean places p implies invpγq invpγ1q.

Define inv : HompGK , Gq Ñ IK by invpπq invppπ|Dpqq.

Then for any finite group G and an admissible invariant inv we define

NinvpK,G;Xq #tπ : GK G| NKQpinvpπqq Xu

If we additionally choose local restrictions Σ pΣpq for Σp HompDp, Gq, we define

NinvpK,Σ;Xq #tπ : GK G| pπ|Dpq P Σ,NKQpinvpπqq Xu

97

3.1 Upper Bounds

We will give upper bounds for the growth of NinvpK,Σ;Xq as X Ñ 8. The weak form

of Malle’s conjecture defines the apGq invariant to be the smallest exponent of p that

can appear in the discriminant of a tamely ramified extension for all but finitely many

places p, so we can make the analogous definition:

Definition 3.2. Fix G a finite group, inv admissible, and Σ pΣpq. Then define

ainvpΣq lim infNKQppqÑ8

minγPΣp,invpγqp1q

νppinvpγqq

If Σ pHompDp, Gqq is trivial, we denote this by ainvpGq.

If G Sn is a transitive subgroup, inv disc is the corresponding discriminant, and

Σp HompDp, Gq for all p P P then ainvpΣq apGq matches the invariant predicted by

Malle.

Using purely group theoretic techniques, we prove an upper bound for NinvpK,Σ;Xqwhen G is solvable which not only gives new evidence for the weak form of Malle’s

conjecture, but it suggests that ainvpΣq plays the same role as apGq for the analogous

counting function.

Theorem 3.3. Fix a normal series

t1u G0 ¤ G1 ¤ ¤ Gm1 Gm G

with Gi G and nilpotent factors GiGi1. Then lim supXÑ8 logpNinvpK,Σ;Xqq logX

is bounded above by

1

ainvpΣq

1

m1

i1

NipEi 1qEi

¸` prime

ν` p|GiGi1|q lim suprL:Ks¤Ni,DLQÑ8

logp|ClpLqr`s|qlogpDLQq

98

where Ni |pGGi1qCGpGiGi1q|, Ei is the order of the largest cyclic group in

pGGi1qCGpGiGi1q, and ν`pnq is the power of ` in the prime factorization of n.

When G is nilpotent we can take m 1 and get an upper bound of 1ainvpΣq. We

can also use the trivial bound |ClpLqr`s| ! D12εLQ to give an explicit bound

lim supXÑ8

logpNpK,Σ;XqqlogX

¤ 1

ainvpΣq

1

m1

i1

NipEi 1q2Ei

Ω p|GiGi1|q

The trivial bound for |ClpLqr`s| has been improved upon by small increments (see the

introduction to Ellenberg-Pierce-Wood [20] for good survey of these results, including

conditional improvements like Ellenberg-Venkatesh [24], unconditional improvements

like Bhargava et. al [7], and improvements on average or for all but an exceptional

family of fields like Ellenberg-Pierce-Wood’s [20] main result), and every improvement

on these bounds gives an improved bound for NinvpK,Σ;Xq. Moreover, it is conjectured

that |ClpLqr`s| ! DεLQ for rL : Ks bounded (as discussed in Brumer-Silverman [12],

Duke [17], Ellenberg-Pierce-Wood [20], and Zhang [61]). This conjecture together with

Theorem 3.3 suggests that

NinvpK,Σ;Xq ! X1ainvpΣqε

This gives new theoretical evidence for the upper bound of Malle’s conjecture for solvable

groups, including the analogous upper bounds for restricted local behavior and ordering

by other admissible invariants.

3.1.1 Group Theoretic Lemmas

Consider an exact sequence of finite groups

1 N G GN 1i q

99

The goal of this section will be to relate |HompH,Gq| to |HompH,Nq| and |HompH,GNq|for any (not necessarily finite) group H.

Motivating Example: Suppose G A is an abelian group. Then HompH,q H1pH,q is a left exact functor on abelain groups with the trivial H action. In other

words, the following is an exact sequence:

0 HompH,Nq HompH,Aq HompH,ANqi q

So in particular

|HompH,Aq| ¤ |HompH,ANq| |HompH,Nq|

We will prove an analogous property for nonabelian groups:

Theorem 3.4. Suppose |HompH,Gq| 8 and fix a normal series

t1u G0 ¤ G1 ¤ ¤ Gm1 ¤ Gm G

with Gi G. Then there exist normal subgroups Mi H for i 1, ...,m such that

HMi ãÑ pGGi1qCGpGiGi1q, Mm H, and

|HompH,Gq| ¤m¹i1

|HMi||GGi1| |HompMi, GiGi1q|

In particular, for any nilpotent group we can choose a composition series which is

a refinement of the upper central series. Being a refinement of the upper central series

implies CGpGiGi1q GGi1 and Mi H for all i 1, ...,m, and being a composition

series implies GiGi1 is cyclic of prime order. If |G| ±`e` is the prime factorization,

then

|HompH,Gq| ¤¹`

|HompH,C`q|e`

For nonnilpotent groups, we will sometimes get M H. We will see in Section 3.1.2

that this corresponds to considering extensions of the base field.

100

Proof of Theorem 3.4. This theorem is proven by induction on the length m of the

normal series. For m 1, HM ãÑ GCGp1q 1 and the result is immediate. For the

inductive step, we require the following lemma:

Lemma 3.5. Suppose |HompH,Gq| 8. Then there exists a normal subgroup M H

such that HM ãÑ GCGpNq and

|HompH,Gq| ¤ |HM ||G| |HompH,GNq| |HompM,Nq|

Given this lemma, we have

|HompH,Gq| ¤ |HM1||G| |HompM1, G1q| |HompH,GG1q|

where GG1 has the corresponding normal series GiG1 of length m 1. We then apply

the inductive hypothesis to conclude the proof.

The proof of Lemma 3.5 will follow from two technical results in group theory. First,

we introduce some notation:

• For α P AutpGq, we will often denote xα αpxq for x P G.

• Given a group action φ : H Ñ G define the set of crossed homomorphisms

Z1φpH,Gq tf : H Ñ G|fpxyq fpxqfpyqφpxqu

• Define the homomorphism κ : G Ñ AutpNq sending x to the conjugation by x

map py ÞÑ yxq for any y P N . N is a normal subgroup of G, so this is well-defined.

(Notice also that ypκgqpxq ygpxq, highlighting the usefulness of this notation for

automorphisms.)

101

• Given any two maps f, g : H Ñ G (not necessarily homomorphisms), define the

map pf gq : H Ñ G by x ÞÑ fpxqgpxq. Similarly for any set B MapspH,Gqdefine f B tf b|b P Bu and B f tb f |b P Bu. The operation makes

MapspH,Gq into a group, but in general HompH,Gq is not a subgroup because it

is not closed.

Lemma 3.6. Suppose g P HompH,Gq and qpgq g under the induced map

HompH,Gq HompH,GNqq

Then the fiber above g is given by

q1 pgq Z1

κgpH,Nq g

Proof. Suppose there exists g P HompH,Gq such that βpgq g. For any f P Z1κgpH,Nq,

it follows that

pf gqpxyq fpxyqgpxyq

fpxqfpyqpκgqpxqgpxqgpyq

fpxqfpyqgpxqgpxqgpyq

fpxqgpxqfpyqgpyq

pf gqpxqpf gqpyq

Clearly qpf gq qpgq g by im f N ker q. Therefore Z1κgpH,Nq g q1

pgq.For the reverse containment, suppose f P HompH,Gq such that qpfq g. Then

102

fpxqgpxq1 P N for every x P H and

pf g1qpxyq fpxyqgpxyq1

fpxqfpyqgpyq1gpxq1

fpxqgpxq1pfpyqgpyq1qgpxq

pf g1qpxqpf g1qpyqgpxq

pf g1qpxqpf g1qpyqpκgqpxq

Therefore f g1 P Z1κgpG, Jq, so that f P Z1

κgpG, Jq g. The opposite containment

qpgq Z1κgpH,Nq g then follows.

Lemma 3.6 gets us part of the way to proving Lemma 3.5. We get the following

bound for all groups G by bounding each fiber above by the size of the largest fiber:

|HompH,Gq| ¤ |HompH,GNq| supgPHompH,Gq

|Z1κgpH,Nq|

Continuing on this path, we want to relate crossed homomorphisms back to homomor-

phisms in a different way.

Lemma 3.7. Let H act on G by the homomorphism φ : H Ñ G. Then the restriction

map f ÞÑ f |kerφ defines a map

α : Z1φpH,Gq Ñ Hompkerφ,Gq

which has fibers of size at most |H kerφ||G|.

Proof. Given any f P Z1φpH,Gq, the restriction f |kerφ belongs to Z1

φpkerφ,Gq. Note that

φ|kerφ : kerφÑ AutpGq is the trivial map, so Z1φpkerφ,Gq Hompkerφ,Gq.

103

Suppose f, g P Z1φpH,Gq such that f |kerφ g|kerφ. Then f g1 is a map sending

kerφ to 1 such that

pf g1qpxyq fpxyqgpxyq1

fpxqfpyqφpxqgpyqφpxqgpxq

fpxqgpxq1pfpyqφpxqgpyqφpxqqgpxq

pf g1qpxqpf g1qpyqpκgqpxqφpxq

For any y P H and z P kerφ, pf g1qpzq 1 implies

pf g1qpyzq pf g1qpyqpf g1qpzqpκgqpyqφpyq

pf g1qpyq

Therefore pf g1q factors through the group of left cosets H kerφ, so that the fiber

α1pf |kerφq embeds in MapspH kerφ,Gq, which has size |H kerφ||G|.

Lemma 3.7 gives us the next step, giving us the following bound for g P HompH,Gq:

|Z1κgpH,Gq| ¤ |H kerpκgq||G| |Hompkerpκgq, Gq|

Putting this together with Lemma 3.6 shows

|HompH,Gq| ¤ |HompH,GNq| supgPHompH,Gq

|H kerpκgq||G||Hompkerpκgq, Nq|

Lemma 3.5 then follows from |HompH,Gq| 8, so there exists some g P HompH,Gqattaining the maximum value. Setting M kerpκgq and noting that H kerpκgq ãÑGCGpNq by CGpNq kerκ concludes the proof.

104

3.1.2 Proof of Theorem 3.3

We will first prove results counting extensions which are unramified outside of a finite

set of places T P . Once we have done this, we will piece these together over all finite

subsets T P to prove Theorem 3.3.

Before we can even use Theorem 3.4, we need to have a group H with |HompH,Gq| 8. For a finite set of places T P , let GT

K be the Galois group of the maximal extension

of K unramified outside of T .

Lemma 3.8. For any finite set of places T P and finite group G, |HompGTK , Gq| 8.

Proof. The decomposition groupsDp are finitely generated, which implies |HompDp, Gq| 8. The inclusions Dp ãÑ GT

K for p P T are well-defined up to conjugation, and they

induce a restriction map on homomorphisms:

res : HompGTK , Gq Ñ

¹pPT

HompDp, Gq

Suppose respfq respgq, then in particular the higher ramification groups have the same

images fpIp,iq gpIp,iq (including i 1). Let Kf be the fixed field of ker f and Kg be

the fixed field of ker g. Then the inertia and ramification indices are equal by the results

for i 1, 0. Also, the map φ : NÑ N given by

φpuq » u

0

dt

rIppKhKq0 : IppKhKqus

is the same for h f or g by all higher ramification groups having the same images in f

105

and g. Moreover, for each p P T we get the following relationship between discriminants:

νppdiscpKfKqq fppKfKq8

i0

p|IppKfKqi| 1q

fppKgKq8

i0

p|IppKfKqφpiq| 1q

fppKgKq8

i0

p|fpIφpiqp q| 1q

fppKgKq8

i0

p|gpIφpiqp q| 1q

fppKgKq8

i0

p|IppKgKqφpiqq| 1q

fppKgKq8

i0

p|IppKgKqi| 1q

νppdiscpKgKqq

Therefore discpKfKq discpKgKq because only p P T is allowed to be ramified. There

are only finitely many fields with a given discriminant, which implies there are finitely

many homomorphisms f : GTK Ñ G such that Kf has a given discriminant. This implies

that the fibers of the restriction map are all finite, and the range is finite. Therefore

HompGTK , Gq is a finite union of finite fibers, which forces it to be finite.

We can now give apply Theorem 3.4 to H GTK to prove the following:

Theorem 3.9. Let G be a finite group with a normal series

t1u G0 ¤ G1 ¤ ¤ Gm1 ¤ Gm G

with Gi G, and T P a finite set of places of K. Then there exists a family of field

extensions Li,T K for i 1, ...,m depending on G and T which are unramified outside

106

of T , Lm,T K, and have GalpLi,T Kq ãÑ pGGi1qCGpGiGi1q such that

|HompGTK , Gq| ¤

m¹i1

|GalpLi,T Kq||GGi1| |HompGT pLi,T qLi,T

, GiGi1q|

where T pLq is the set of places of L lying above a place p P T .

Proof. Setting H GTK in Theorem 3.4 gives a family of normal subgroups Mi GT

K

for i 1, ...,m such that GTKMi ãÑ pGGi1qCGpGiGi1q and

|HompGTK , Gq| ¤

m¹i1

|GTKMi||GGi1| |HompMi, GiGi1q|

By the Galois correspondence, each Mi GTK corresponds to a Galois extension Li,T K

unramified away from T where Mi is the Galois group of the maximal extension of

Li,T unramified away from T and GTKMi GalpLi,T Kq. Therefore Mi G

T pLi,T qLi,T

,

GTKMi GalpLi,T Kq, and Mm GT

K implies Lm,T K concluding the proof.

Note that we can make the following bound depending only on pGiqi:m¹i1

|GTKMi||GGi1| ¤

m¹i1

|pGGi1qCGpGiGi1q||GGi1|

Call this constant CppGiqiq.We can now prove Theorem 3.3 by employing some class field theory.

Proof of Theorem 3.3. Consider the Dirichlet series

LinvpK,Σ, sq ¸aPIK

|tπ : GL G|pπ|Dpq P Σ, invpπq au|NKQpaqs

Then NinvpK,Σ;Xq is the sum of the coefficients for all NKQpaq X, so it suffices to

study where this function absolutely converges. Denote

ainvpp,Σq minγPΣ

νppinvpγqq

107

so that ainvpΣq lim inf ainvpp,Σq. Then

|LinvpK,Σ, sq| ¸aPIK

|tπ : GL G|pπ|Dpq P Σ, invpπq au|NKQpaq<psq

¤¸TP

|tπ : GL G|pπ|Dpq P Σ, πpIpq 1 if p R T u|

¹

pPTNKQppqainvpp,Σq

<psq

¤¸TP

|HompGTK , Gq|

¹pPTNKQppqainvpp,Σq

<psq

Theorem 3.9 then implies for any normal series pGiqmi0 that

|LinvpK,Σ, sq| ¤ CppGiqiq¸TP

m¹i1

|HompGT pLi,T qLi,T

, GiGi1q|¹

pPTNKQppqainvpp,Σq

<psq

We strategically chose pGiqi to have nilpotent factors. If we apply Theorem 3.4 to the

upper central series of GiGi1 then given the prime factorization |GiGi1| ±`e`,i it

follows that

|HompGT pLi,T qLi,T

, GiGi1q| ¤¹`

|HompGT pLi,T qLi,T

, C`q|e`,i

We will use class field theory to bound |HompGTLpLq, C`q| for L ramified only in T . There

is an exact sequence for each L:

±PPT pLq IPpLabLq pGT pLq

L qab ClpLq

The inflation and restriction maps then produce an exact sequence:

HompClpLq, C`q HompGT pLqL , C`q Homp±PPT pLq IPpLabLq, C`q

Therefore

|HompGT pLqL , C`q| ¤ |HompClpLq, C`q|

¹PPT pLq

|HompIPpLabLq, C`q|

108

For the inertia factors we have two cases. If P - ` then HompIP , C`q factors through

tame ramification, which is cyclic. Therefore

|HompIPpLabLq, C`q| ¤ `

If P | `, then there can be wild ramification. Local class field theory tells us that

IPpLabLq cyclic ZrL:Qs` so that

|HompIPpLabLq, C`q| ¤ `rL:Qs1

Noting that there are at most Ni |pGGi1qCGpGiGi1q| ¥ |GalpLT pLi,T qi,T Kq| places

P P T pLi,T q above each p P T it follows that¹PPT pLq

|HompIPpLabLq, Cn` q| ¤ p`NiprL:Qs1qq#T

For the class group factor, define

a`,i lim suprL:Ks¤Ni,DLQÑ8

logp|ClpLqr`s|qlogpDLQq

so that |ClpLqr`s| ! Da`,iεLQ . Noting that Lm,T K, it follows that there exists a constant

c`,ipεq such that

|HompClpLi,T q, C`q| ¤

$''&''%c`,ipεqNKQpDLi,T Kqa`,iε i m

c`,ipεq i m

Putting this information together, there exist positive constants Cpεq ±mi1

±` c`,ipεqe`,i

and M ±mi1

±` `e`,iNiprLi,T :Ks1q such that

m¹i1

|HompGT pLi,T qLi,T

, GiGi1q| ¤m¹i1

¹`

|HompGT pLi,T qLi,T

, C`q|e`,i

¤ CpεqpMq#Tm1¹i1

¹`

NKQpdiscpLi,T Kqqe`,ia`,iε

¤ CpεqpMq#T¹pPTNKQppq

°m1i1

°` e`,ia`,iνppDLi,T Kqε

109

We remark that each extension Li,T K is tamely ramified at all but finitely many places

p (namely those for which p | |G|), and because GalpLi,T Kq ¤ pGGi1qCGpGiGi1qqit follows that whenever p is at most tamely ramified

νppDLiKq ¤ maxxPpGGi1CGpGiGi1qq

Ni

1 1

|xxy|

¤ Ni

1 1

Ei

Thus for all but finitely many places

m1

i1

¸`

e`,ia`,iνppDLi,T Kq ¤m1

i1

¸`

NipEi 1qEi

e`,ia`,i

Only finitely many places are allowed to be wildly ramified, so at the cost of enlarging

Cpεq it follows that

m1¹i0

¹`

|HompGT pLi,T qLi,T

, C`q|e`,i ¤ CpεqpMq#T¹pPTNKQppq

°m1i1

°`

NipEi1q

Eie`,ia`,iε

Call the exponent on the right side b ε. Putting this information back into the bounds

of our L-function, we find

|LinvpK,Σ, sq| ¤ CppGiqiqCpεq¸TP

M#T

¹pPTNKQppqainvpp,Σq<psqbε

CppGiqiqCpεq¹pPP

1MNKQppqainvpp,Σq<psqbε

This absolutely converges whenever

1 ¡ lim supNKQppqÑ8

ainvpp,Σq<psq b ε

ainvpΣq<psq b ε

<psq ¡ 1

ainvpΣqpb 1 εq

110

For an appropriate choice of ε, we then apply a Mellin transform to show that

NinvpK,Σ;Xq ! Xb1

ainvpΣqε

We conclude the proof by plugging back in for b as follows:

lim supXÑ8

logpNinvpK,Σ;XqqlogX

¤ 1

ainvpΣq

1

m1

i1

¸`

NipEi 1qEi

e`,ia`,i

where e`,i ν`p|GiGi1|q and a`,i lim suprL:Ks¤Ni

logp|ClpLqr`s|q logpDLQq.

3.1.3 Data

In this section we provide data for the logarithmic growth of NpK,G;Xq when G Sn

is a transitive, solvable subgroup and n small. We will directly compare the bounds

given by Theorem 3.3 and Minkowski’s bounds on the class number to the bounds due

to Dummit [19] and Schmidt [52].

Some families of groups are easy to produce bounds for NpK,G;Xq using compu-

tations done by hand, such as Dn Sn as discussed in the beginning of Section 3.1.

In general though, we can get a more complete picture by using a computer algebra

program. All of the computations in this section are done using MAGMA.

One of the computational drawbacks of Dummit’s result is the computational power

necessary to compute sets of primary invariants. Dummit’s data extends to transitive

groups of degree 8 and then covers only four transitive groups of degree 9 because of the

length of time computations were taking. If we apply the trivial bound from Minkowski

to Theorem 3.3 the bulk of the computations are done computing a normal series for G

with nilpotent factors, which MAGMA is able perform very quickly.

We compute the normal series by choosing a minimal normal subgroup N1 G, then

choosing a minimal normal subgroup N2N1 GN1, and iterating until GNm 1.

111

We do not optimize our choice of Ni, so it is possible that some of these bounds could

be improved by choosing a different normal series. For the most part, groups of small

order have very few minimal normal subgroups and changing our choice will not change

the bounds listed in this section.

We will use nTd to denote the group TranstiveGroup(n,d) in MAGMA’s database,

and we will only include solvable groups. In each column, we will give the corresponding

upper bound to lim supXÑ8 logpNpK,G;Xqq logX: the “Result” column will have the

upper bound from Theorem 3.3 combined with the trivial bound on `-torsion of the

class group from Minkowski’s Theorem, the “Malle” column will have the predicted

upper bound from the weak form of Malle’s conjecture, the “Dummit/Q” column will

have Dummit’s bound for extensions over Q, and the “Schmidt” column will have the

n24

bound from Schmidt’s Theorem. We remark that Dummit’s bounds depend slightly

on the field K, if a is the bound given over Q then a11rK : Qs is the corresponding

bound over K. We will put a on nilpotent groups, for which the bounds from Theorem

3.3 provably agree with Malle’s predicted bounds in all degrees.

degree 5 Isom. to Result Malle Dummit/Q Schmidt

5T1 C5 1/4 1/4 11/8 7/4

5T2 D5 3/4 1/2 11/8 7/4

5T3 F20 5/4 1/2 13/8 7/4

Bhargava-Shankar-Wang [8] showed that 1 is an upper bound for all degree 5 exten-

sions.

112

degree 6 Isom. to Result Malle Dummit/Q Schmidt

6T1 C6 1/3 1/3 7/3 2

6T2 S3 1/2 1/3 11/6 2

6T3 S3 C2 3/4 1/2 7/3 2

6T4 A4 3/2 1/2 2 2

6T5 F18 3/4 1/2 7/4 2

6T6 A4 C2 3 1 8/3 2

6T7 S4 11/4 1/2 13/6 2

6T8 S4 11/4 1/2 8/3 2

6T9 S3 S3 1 1/2 2 2

6T10 F36 2 1/2 17/8 2

6T11 S4 C2 11/2 1 8/3 2

6T13 F36 C2 7 1 2 2

We remark that the strong form of Malle’s conjecture has been verified for 6T2 over

Q by Bhargava-Wood [9] and for 6T6 and 6T11 by Kluners [36]. Therefore these realize

the upper bounds predicted by Malle.

degree 7 Isom. to Result Malle Dummit/Q Schmidt

7T1 C7 1/6 1/6 19/12 9/4

7T2 D7 1/2 1/3 19/12 9/4

7T3 F21 1/2 1/4 7/4 9/4

7T4 F42 7/6 1/3 2 9/4

113

degree 8 Isom. to Result Malle Dummit/Q Schmidt

8T1 C8 1/4 1/4 11/4 5/2

8T2 C4 C2 1/4 1/4 17/8 5/2

8T3 C32 1/4 1/4 13/8 5/2

8T4 D4 1/4 1/4 17/8 5/2

8T5 Q8 1/4 1/4 19/8 5/2

8T6 1/3 1/3 11/4 5/2

8T7 1/2 1/2 3 5/2

8T8 1/3 1/3 3 5/2

8T9 D4 C2 1/2 1/2 17/8 5/2

8T10 1/2 1/2 9/4 5/2

8T11 1/2 1/2 19/8 5/2

8T12 SL2pF3q 3/4 1/4 23/8 5/2

8T13 A4 C2 3/4 1/4 21/8 5/2

8T14 S4 11/8 1/4 11/4 5/2

8T15 1/2 1/2 3 5/2

8T16 1/2 1/2 3 5/2

8T17 1/2 1/2 3 5/2

8T18 1/2 1/2 9/4 5/2

8T19 1/2 1/2 5/2 5/2

8T20 1/2 1/2 5/2 5/2

8T21 1/2 1/2 19/8 5/2

8T22 1/2 1/2 19/8 5/2

114

8T23 GL2pF3q 11/6 1/3 27/8 5/2

8T24 S4 C2 11/4 1/2 11/4 5/2

8T25 F56 5/2 1/4 27/14 5/2

8T26 1/2 1/2 3 5/2

8T27 1 1 3 5/2

8T28 1/2 1/2 3 5/2

8T29 1/2 1/2 5/2 5/2

8T30 1/2 1/2 3 5/2

8T31 1 1 19/8 5/2

8T32 5/2 1/2 23/8 5/2

8T33 C22 C6 5/2 1/2 2 5/2

8T34 E24 D6 19/4 1/2 2 5/2

8T35 1 1 3 5/2

8T36 C32 F21 29/4 1/4 15/7 5/2

8T38 5 1 27/8 5/2

8T39 19/4 1/2 23/8 5/2

8T40 19/4 1/2 27/8 5/2

8T41 C32 S4 19/4 1/2 2 5/2

8T42 63/4 1/2 13/6 5/2

8T44 19/2 1 27/8 5/2

8T45 31 1/2 7/3 5/2

8T46 29 1/2 7/3 5/2

8T47 127 1/2 7/3 5/2

115

We remark that the strong form of Malle’s conjecture has been verified for 8T27,

8T31, 8T35, 8T38, and 8T44 by Kluners [36]. The last two are nonnilpotent and Kluners’

result shows that they still realize the upper bounds predicted by Malle.

degree 9 Isom. to Result Malle Dummit/Q Schmidt

9T1 C9 1/6 1/6 11/4

9T2 C3 C3 1/6 1/6 11/4

9T3 D9 1/2 1/4 13/6 11/4

9T4 S3 C3 1/2 1/3 23/12 11/4

9T5 C23 C2 1/2 1/4 19/12 11/4

9T6 1/4 1/4 11/4

9T7 1/4 1/4 11/4

9T8 S3 S3 2/3 1/3 2 11/4

9T9 1 1/4 11/4

9T10 1/2 1/4 11/4

9T11 1/2 1/4 11/4

9T12 2/3 1/3 11/4

9T13 2/3 1/3 11/4

9T14 7/4 1/4 11/4

9T15 2 1/4 11/4

9T16 7/3 1/3 11/4

9T17 1/2 1/2 11/4

9T18 5/6 1/3 11/4

9T19 5 1/3 11/4

116

9T20 1 1/2 11/4

9T21 5/4 1/2 11/4

9T22 5/4 1/2 11/4

9T23 23/4 1/4 11/4

9T24 3/2 1/2 11/4

9T25 15/2 1/2 11/4

9T26 95/6 1/3 11/4

9T28 33 1 11/4

9T29 65/4 1/2 11/4

9T30 65/4 1/2 11/4

9T31 131/2 1 11/4

We only provide Dummit’s bounds for the groups which Dummit computed in [19].

Dummit’s Theorem does give bounds for all proper transitive subgroups G Sn, but

it becomes computationally intensive to find a set of primary invariants in order to

compute the bound. We additionally remark that Wang [56] proved that 9T4 satisfies

the strong form of Malle’s conjecture, so in particular has an upper bound of 13.

117

Appendix A

The Quaternion Group

In this appendix we detail the remaining cases in the proof of Theorem 2.23 from Section

2.2.2.

A.1 The Asymptotic Count: Remaining Imaginary

Cases

We presented the case d 0 with d 1 mod 4 in Section 2.2.2 of the main body of the

paper. There are two imaginary cases remaining, d 0 with d 0 or 4 mod 8.

A.1.1 The case d 0, d 4 mod 8

If we include degenerate factorizations in Lemmermeyer’s classification of unramified

H8-extensions, then it follows for a quadratic field K of discriminant d

|HomσpGalpKurKq, H8q| ¸

d4d1d2d3

8ap4d1, d2, d3q,

where the sum is over factorizations into odd, squarefree integers with either 4d1, d2, d3

or 4d1,d2, d3 discriminants (Note the factor of 8 is to account for automorphisms of

118

H8 which lift to G1. |HomσpGalpKurKq, H8q| is expanded as follows:

1

2

¸d4d1d2d3

1

d2d3

2

¹p|d3

1

d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

d3d1

p

¸

d4d1d2d3

1

d2d3

2

¹p|d3

1

d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

d3d1

p

¸

d4±Di

1

D1D2D4D5

2

1

m0

1

2m

1

D2D4

p1mq

1

D0D4

¹i,j

Di

Dj

Φpi,jq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3 p1qm1 mod 4, D2D5 p1qm1 mod 4, and D1D4 1 mod 4. This is the

exact set-up needed to apply Theorem 1.6.

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

119

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| ¸

d4±Ei

1

E2E3

2

1

m0

1

2m

1

E2

p1mq

1

E0

¹i,j

EiEj

Ψpi,jq,

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1, E2E3 p1qm1 mod 4.

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq1

We can multiply these together to get an expression for |HomσpGalpKurKq, Hj18 Cj2

4 Cj3

2 q|.We must get this back into the form in Theorem 1.6. As in Section 2.2.2 set Vj1,j2

t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand |HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q| as follows:

2pωpdq1qj3¸

d±Du

¸Ct1,...,j1j2u

¸J1t1,...,j1u

¸J2tj11,...,j1j2u

2|J1||J2|χC,J1,J2

j1,j2ppDuquPV q

¹u,v

Du

Dv

Φj1,j2pu,vq

,

120

where the character is given by

χC,J1,J2

j1,j2ppDuquPV q

¹`PC,`¤j1

±u`1,2,4,5Du

2

¹`PJ1

1±

u`2,4Du

¹`RJ1,`¤j1

1±

u`0,4Du

¹

`PC,`¡j1

±u`2,5Du

2

¹`PJ2

1±u`2Du

¹`RJ2,`¡j1

1±u`0Du

,

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du

$''&''%p1qχJ1

p`q1 mod 4 1 ¤ ` ¤ j1, i 0, 3, 2, 5

1 mod 4 1 ¤ ` ¤ j1, i 1, 4¹u`i,i1

Du p1qχJ2p`q1 mod 4 j1 1 ¤ ` ¤ j1 j2

(Note: χA is the characteristic function of a set A) For each partition k j1 j2 j3

and tuple pC, J1, J2q we can now apply Theorem 1.6 with M 8 and T ppZ8ZqqV

describing the above congruence conditions to compute

¸d X,d4 mod 8

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

The function Φj1j2 Ψj1j2 is indentical to the one found in Section 2.2.2 for the

case d 0 and d 1 mod 4. Thus the unlinked sets of maximal size have already been

classified to be of size 3j12j2 by Proposition 2.26. Therefore Theorem 1.6 implies the

main term only comes from j1 k, j2 j3 0, in which case Proposition 2.24 shows

that the maximal unlinked sets are exactly sets of the form U tu : ui P siu for a type

s P S tA,Buk. Therefore we can take j2, j3 0 and J2 H.

121

Thus Theorem 1.6 tells us that

¸d X,d4 mod 8

|HomσpGalpKurKq, Hk8 q|

¸Ct1,...,j1j2u

¸J1t1,...,j1u

2|J1|

¸UγC,J1pUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

2

3k

¸U

¸Ct1,...,j1j2u

¸J1t1,...,j1u

2|J1|

¸UγC,J1pUq

¸d X,d4 mod 8

3kωpdq

OXplogXq3k2ε

The only remaining step is to compute the leading constant, where

γC,J1pUq 43k¸

phuqPT pUqχC,J1,Hk,0 pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies

¹u`i

tu

$''&''%χJ1p`q 1 mod 4 i 0, 3, 2, 5

1 mod 4 i 1, 4

As in Section 2.2.2, define the matrix Mk recursively. Then the y P F3k

2 satisfying the

above condition (written additively) for an independent set U of type s P S are solutions

of

¸u:uij

yu

$''&''%p1qχJ1

piq1 if j 0, 3, 2, 5 and j P si

0 else

By construction

Mky ¸

u:uijyu

pi,jq

122

which makes these conditions equivalent to Mky w for an appropriate w P F3k2 . This

set of solutions is the coset y kerMk.

Lemma A.1. For all k ¥ 1

¸UγC,J1 pUq 3k22k1.

Proof. Recall

χC,J1,Hk,0 pphuquPV q

¹`PC,`¤j1

±u`1,2,4,5 hu

2

¹`PJ1

1±

u`2,4 hu

¹`RJ1,`¤j1

1±

u`0,4 hu

k¹`1

p1qχCp`q°

u`1,2,4,5h2u1

8χJ1

p`q°u`2,4hu1

2p1χJ1

p`qq°u`0,4hu1

2

The conditions on T are given modulo 4, so we can take a sum over all equivalence class

of phuq P ppZ8Zqq|V | mod 4. Notice that

p5xq2 1

8 25x2 1

8 x2 x2 1

8 1 x2 1

8mod 2

for any odd x. Therefore relating hu12

xu P F2 we find that

¸pxuq mod 4

χC,J1,Hk,0 ppxuquPV q

k¹`1

p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

¸GU

k¹`1

p1qχCp`qp°

u`1,2,4,5 xuχGpuqq

k¹`1

p1qχCp`q°

u`1,2,4,5 xuχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

1 p1qχCp`q3k

$''&''%23k

±k`1p1qχJ1

p`q°u`2,4 xup1χJ1p`qq°u`0,4 xu C H

0 else

123

Therefore we can write¸C

¸J1

¸UγC,J1pUq 23k

¸sPS

¸xPykerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

¸J1

2|J1|k¹`1

¹uPUs

p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

23k¸sPS

¸xPykerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

¸J1

2|J1|k¹`1

p1q°

uPUsχJ1

p`qχt2,4upu`qxup1χJ1p`qqχt0,4upu`qxu

23k¸

xPykerMk

¸J1

2|J1|

k¹`1

1

¹tu,vu

p1q°

uPUBΦpui,viqxuxvχJ1

p`qχt2,4upu`qxup1χJ1p`qqχt0,4upu`qxu

(Note: we get rid of the sum over types s P S by using the binomial theorem)

Notice that for all j 1, ..., k we have¸tu,vuUB

Φpuj, vjqxuxv ¸

uj0,vj2

xuxv ¸

uj0,vj4

xuxv ¸

uj2,vj4

xuxv

¸uj0

xu¸vj2

xv ¸uj0

xu¸vj4

xv ¸uj2

xu¸vj4

xv

and

¸uPUB

χJ1pjqχt2,4upujqxu p1 χJ1pjqqχt0,4upjqxu

$''&''%°uj2 xu

°uj4 xu j P J1°

uj2 xu °uj4 xu j R J1

Recall that°u`m xu χJ1p`q 1 if m 0, 3, 2, 5 and 0 if m 1, 4. Then it follows

that ¸tu,vuUB

Φpuj, vjqxuxv pχJ1pjq 1q2 χJ1pjq 1 mod 2

¸uPUB

χJ1pjqχt2,4upujqxu p1 χJ1pjqqχt0,4upjqxu χJ1pjq 1 mod 2

124

Adding these together is trivial, which implies

¸C

¸J1

¸UγC,J1pUq 23k

¸xPykerMk

¸J1t1,...,ku

2|J1|2k

23kkdim kerMkp32qk

Recall Lemma 2.29 from the main body of the paper showing that dim kerMk 3k 2k 1, concluding the proof.

In conclusion, plugging in the value of this constant yields the result from Theorem

2.23:

¸d X,d4 mod 8

|SurjσpGalpKurKq, Hk8 q|

1

4k

¸d X,d4 mod 8

3kωpdq OXplogXq3k2ε

A.1.2 The case d 0, d 0 mod 8

If we include degenerate factorizations in Lemmermeyer’s classification of unramified

H8-extensions, then it follows for a quadratic field K of discriminant d

|HomσpGalpKurKq, H8q| ¸

d4d1d2d3

8ap8d1, d2, d3q,

where the sum is over factorizations into odd, squarefree integers with either p8d1, d2, d3q,p8d1, d2, d3q, or p8d1,d2, d3q are discriminants (Note the factor of 8 is to account for

125

automorphisms of H8 which lift to G1. |HomσpGalpKurKq, H8q| is expanded as follows:

1

2

¸d8d1d2d3

1

d2d3

2

¹p|d3

1

2d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

2d3d1

p

¸

d8d1d2d3

1

d2d3

2

¹p|d3

1

2d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

2d3d1

p

¸

d8D0D1D2D3D4D5

1

D1D2D4D5

2

2

D2D4

1

m0

1

2

1

D2D4

m

1

D0D4

p1mq

¹tu,vu

Du

Dv

Φpu,vq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3 1 mod 4, D2D5 p1qm1 mod 4, and D1D4 1 mod 4. This is the exact

set-up needed to apply Theorem 1.6.

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

126

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| ¸

d8±Ei

1

E2E3

2

2

E2

1

m0

1

2m

1

E2

p1mq

1

E0

¹i,j

EiEj

Ψpi,jq,

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1 1 mod 4 and E2E3 p1qm1 mod 4.

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq1

We can multiply these together to get an expression for |HomσpGalpKurKq, Hj18 Cj2

4 Cj3

2 q|.We must get this back into the form in Theorem 1.6. As in Section 2.2.2 set Vj1,j2

t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand |HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q| as follows:

2pωpdq1qj3¸

d±Du

¸Ct1,...,j1j2u

¸J1t1,...,j1u

¸J2tj11,...,j1j2u

2|J1||J2|

χC,J1,J2

j1,j2ppDuquPV q

¹u,v

Du

Dv

Φj1,j2pu,vq

,

127

where the character is given by

χC,J1,J2

j1,j2ppDuquPV q

¹`PC,`¤j1

±u`1,2,4,5Du

2

¹`¤j1

2±

u`2,4Du

¹`PJ1

1±

u`2,4Du

¹`RJ1,`¤j1

1±

u`0,4Du

¹

`PC,`¡j1

±u`2,3Du

2

¹`¡j1

2±

u`2Du

¹`PJ2

1±u`2Du

¹`RJ2,`¡j1

1±u`0Du

,

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du

$''''''&''''''%

±uji,i3Du 1 ¤ `, j ¤ j1, i 0, 3

p1qχJ1p`q1 mod 4 1 ¤ ` ¤ j1, i 2, 5

1 mod 4 1 ¤ ` ¤ j1, i 1, 4

¹u`i,i1

Du

$''&''%±

uji,i1Du j1 1 ¤ `, j ¤ j1 j2, i 0, 1

p1qχJ2p`q1 mod 4 j1 1 ¤ ` ¤ j1 j2, i 2, 3

For each partition k j1 j2 j3 and tuple pC, J1, J2q we can now apply Theorem

1.6 with M 8 and T ppZ8ZqqV describing the above congruence conditions to

compute ¸d X,d4 mod 8

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

The function Φj1j2 Ψj1j2 is indentical to the one found in Section 2.2.2 for the

case d 0 and d 1 mod 4. Thus the unlinked sets of maximal size have already been

128

classified to be of size 3j12j2 by Proposition 2.26. Therefore Theorem 1.6 implies the

main term only comes from j1 k, j2 j3 0, in which case Proposition 2.24 shows

that the maximal unlinked sets are exactly sets of the form U tu : ui P siu for a type

s P S tA,Buk. Therefore we can take j2, j3 0 and J2 H.

Thus Theorem 1.6 tells us that

¸d X,d4 mod 8

|HomσpGalpKurKq, Hk8 q|

¸Ct1,...,j1j2u

¸J1t1,...,j1u

2|J1|

¸UγC,J1pUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

1

3k

¸U

¸Ct1,...,j1j2u

¸J1t1,...,j1u

2|J1|

¸UγC,J1pUq

¸d X,d0 mod 8

3kωpdq

OXplogXq3k2ε

The only remaining step is to compute the leading constant, where

γC,J1pUq 43k¸

phuqPT pUqχC,J1,Hk,0 pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies

¹u`i

tu

$''&''%p1qχJ1

p`q1 mod 4 i 2, 5

1 mod 4 i 1, 4

As in Section 2.2.2, define the matrix Mk recursively. Then the y P F3k

2 the above

129

condition (written additively) for an independent set U of type s P S are solutions of

¸u:uij

yu

$''''''&''''''%

°u:u`j yu j 0, 3 and j P si, s`

χJ1piq 1 if j 2, 5 and j P si

0 else

By construction

Mky ¸

u:uijyu

pi,jq

which makes these conditions equivalent to Mky w for 2 different w P F3k2 (One

for each possible value of°u`0,3 yu). This set of solutions is the union of the two cosets

y kerMk.

Lemma A.2. For all k ¥ 1 ¸UγC,J1 pUq 3k22k.

Proof. Recall

χC,J1,Hk,0 pphuquPV q

¹`PC,`¤j1

±u`1,2,4,5 hu

2

2±

u`2,4 hu

¹`PJ1

1±

u`2,4 hu

¹`RJ1,`¤j1

1±

u`0,4 hu

k¹`1

p1qχCp`q°

u`1,2,4,5h2u1

8°u`2,4

h2u1

8

p1qχJ1p`q°u`2,4

hu12

p1χJ1p`qq°u`0,4

hu12

The conditions on T are given modulo 4, so we can take a sum over all equivalence class

of phuq P ppZ8Zqq|V | mod 4. Notice that

p5xq2 1

8 25x2 1

8 x2 x2 1

8 1 x2 1

8mod 2

130

for any odd x. Therefore relating hu12

xu P F2 we find that

¸pxuq mod 4

χC,J1,Hk,0 ppxuquPV q

k¹`1

p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

¸GU

k¹`1

p1qχCp`q°

u`1,2,4,5 χGpuq°

u`2,4 χGpuq

k¹`1

p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

¹uPUs

1 p1qχCp`qχt1,2,4,5upu`qχt2,4upu`q

$''''''&''''''%

23k±k

`1p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

if C ti : si Bu

0 otherwise

Therefore we can write

¸C

¸J1

¸UγC,J1pUq 23k

¸sPS

¸y

¸xPykerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

¸J1

2|J1|k¹`1

¹uPUs

p1qχJ1p`q°u`2,4 xup1χJ1

p`qq°u`0,4 xu

This is equal to the sum over the 2 choices of coset y kerMk of the°γC,J1pUq found

in the d 0, d 4 mod 8 case. Lemma A.1 shows that this is 3k22k1, so multiplying

by 2 concludes the proof.

In conclusion, plugging in the value of this constant yields the result from Theorem

2.23:

¸d X,d0 mod 8

|SurjσpGalpKurKq, Hk8 q|

1

4k

¸d X,d0 mod 8

3kωpdq OXplogXq3k2ε

131

A.2 The Asymptotic Count: Real Cases

When using Lemmermeyer’s classification over real quadratic fields, we have to address

ramification at infinity separately. The classification is for H8-extensions unramified at

all finite places, i.e.

apd1, d2, d3q #tKQp?dq : GalpKQp

?dqq H8, unramified at all finite placesu.

Lemmermeyer [40] does give a classification for the ramification at infinity as well, but

that will not be necessary for our purposes. The following lemma shows that all of the

nuance for the prime infinity only affects the error term:

Lemma A.3. Let

bpd1, d2, d3q #tKQp?dq : GalpKQp

?dqq H8, unramified at all placesu

If Theorem 2.23 holds for apd1, d2, d3q for some leading constant, then

¸d1d2d3 X

bpd1, d2, d3qk 1

2k

¸d1d2d3 X

apd1, d2, d3qk

Proof. For a fixed factorization, let L Qp?d1,?d2,

?d3q so that if KQp?dq is an

H8-extension unramified at all finite places, then L ¤ K and in fact K Lp?µq. d ¡ 0

and Lemmermeyer’s conditions imply di ¡ 0, so define a map

φ : xδ | d : δ is a quadratic discriminantyxd1, d2, d3y Ñ t1,1u

by sending δ ÞÑ δ|δ|. φ is surjective if and only if there exists p 3 mod 4 such

that p | d. Fix that prime, then exactly one of KL and Lp?pµqL is unramified

at 8. This implies that there is a one-to-one correspondence between tKQp?dq :

132

GalpKQp?dqq H8, unramified at all placesu and p kerφ. Lemmemeyer’s classifica-

tion shows that apd1, d2, d3q | kerφ| 2ωpdq3, so because ppq2 1 in the group, we

have bpd1, d2, d3q 12apd1, d2, d3q.

If p | d implies p 1, 2 mod 4, then bpd1, d2, d3q ¤ apd1, d2, d3q.By Landau’s Theorem and Holder’s inequality with 1b 1c 1 it follows that

¸dd1d2d3 X,p|dñp1mod4

apd1, d2, d3qk ¸d X,p|dñp1mod4

1

1b ¸d1d2d3 X

apd1, d2, d3q1c

!

X?logX

1b XplogXq3k1

1c

If we choose b 3k 12 then this is OpXplogXq3k2εq. Therefore

¸d1d2d3 X

bpd1, d2, d3qk 1

2k

¸d1d2d3 X

apd1, d2, d3qk

¤

1 1

2k

¸dd1d2d3 X,p|dñp1mod4

apd1, d2, d3qk

OpXplogXq3k2εq

Applying Theorem 2.23 concludes the proof.

Applying Lemma A.3 lets us address the real cases in much the same way as the

imaginary cases.

A.2.1 The case d ¡ 0, d 1 mod 4

If we include degenerate factorizations in Lemmermeyer’s classification of unramified H8-

extensions and apply Lemma A.3, then it follows for a quadratic field K of discriminant

d

|HomσpGalpKurKq, H8q| ¸

dd1d2d3

81

2apd1, d2, d3q,

133

where the sum is over factorizations into odd, squarefree integers with d1, d2, d3 quadratic

discriminants (Note the factor of 8 is to account for automorphisms of H8 which lift to

G1. |HomσpGalpKurKq, H8q| is expanded as follows:

1

12

¸dd1d2d3

¹p|d3

1

d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

d3d1

p

1

12

¸dD0D1D2D3D4D5

¹tu,vu

Du

Dv

Φpu,vq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3, D2D5, D1D4 1 mod 4. This is the exact set-up needed to apply Theorem 1.6.

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| 1

4

¸d4

±Ei

¹i,j

EiEj

Ψpi,jq,

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1, E2E3 1 mod 4.

134

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq2

We can multiply these together to get an expression for |HomσpGalpKurKq, Hj18 Cj2

4 Cj3

2 q|.We must get this back into the form in Theorem 1.6. As in Section 2.2.2 set Vj1,j2

t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand |HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q| as follows:

3j12pωpdq1qj32j1j2¸

d±Du

χj1,j2ppDuquPV q¹u,v

Du

Dv

Φj1,j2pu,vq

,

where the character is given by

χj1,j2ppDuquPV q 1

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du 1 mod 4 1 ¤ ` ¤ j1

¹u`i,i1

Du 1 mod 4 j1 1 ¤ ` ¤ j1 j2

135

For each partition k j1 j2 j3 and set C we can now apply Theorem 1.6 with M 8

and T ppZ8ZqqV describing the above congruence conditions to compute

¸d X,d4 mod 8

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

The function Φj1j2 Ψj1j2 is indentical to the one found in Section 2.2.2 for the

case d 0 and d 1 mod 4. Thus the unlinked sets of maximal size have already been

classified to be of size 3j12j2 by Proposition 2.26. Therefore Theorem 1.6 implies the

main term only comes from j1 k, j2 j3 0, in which case Proposition 2.24 shows

that the maximal unlinked sets are exactly sets of the form U tu : ui P siu for a type

s P S tA,Buk. Therefore we can take j2, j3 0.

Thus Theorem 1.6 tells us that

¸d X,d1 mod 4

|HomσpGalpKurKq, Hk8 q|

1

12k

¸UγpUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

2

3k22k

¸UγpUq

¸

d X,d1 mod 4

3kωpdq

OXplogXq3k2ε

The only remaining step is to compute the leading constant, where

γpUq 43k¸

phuqPT pUqχk,0pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies ¹u`i

tu 1 mod 4

136

As in Section 2.2.2, define the matrix Mk recursively. Then the y P F3k

2 satisfying

the above condition for an independent set U of type s P S are solutions of¸u:uij

yu 0

By construction

Mky ¸

u:uijyu

pi,jq

which makes these conditions equivalent to Mky 0. This set of solutions is the kernel

kerMk.

Lemma A.4. For all k ¥ 1 ¸UγC pUq 2k1.

Proof. Recall χk,0 1. The conditions on T are given modulo 4, so we can take a sum

over all equivalence class of phuq P ppZ8Zqq|V | mod 4. Therefore relating hu12

xu PF2 we find that ¸

pxuq mod 4

χCk,0ppxuquPV q 23k

Therefore we can write¸C

¸UγCpUq 23k

¸sPS

¸xPkerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

23k¸

xPkerMk

k¹`1

1 p1q

°u,vPUB

Φpu`,v`qxuxv

(Note: we get rid of the sum over types s P S by using the binomial theorem)

Notice that for all j 1, ..., k we have¸tu,vuUB

Φpuj, vjqxuxv ¸

uj0,vj2

xuxv ¸

uj0,vj4

xuxv ¸

uj2,vj4

xuxv

¸uj0

xu¸vj2

xv ¸uj0

xu¸vj4

xv ¸uj2

xu¸vj4

xv

137

Recall that°u`m xu 0. Then it follows that

¸tu,vuUB

Φpuj, vjqxuxv 0

This implies

¸UγpUq 23k

¸xPkerMk

2k

23kdim kerMkk

Recall Lemma 2.29 from the main body of the paper showing that dim kerMk 3k 2k 1, concluding the proof.

In conclusion, plugging in the value of this constant yields the result from Theorem

2.23:

¸d X,d4 mod 8

|SurjσpGalpKurKq, Hk8 q|

1

24k

¸d X,d4 mod 8

3kωpdq OXplogXq3k2ε

A.2.2 The case d ¡ 0, d 4 mod 8

If we include degenerate factorizations in Lemmermeyer’s classification of unramified H8-

extensions and apply Lemma A.3, then it follows for a quadratic field K of discriminant

d

|HomσpGalpKurKq, H8q| ¸

d4d1d2d3

81

2ap4d1, d2, d3q,

where the sum is over factorizations into odd, squarefree integers with 4d1, d2, d3 quadratic

discriminants (Note the factor of 8 is to account for automorphisms of H8 which lift to

138

G1). |HomσpGalpKurKq, H8q| is expanded as follows:

1

4

¸d4d1d2d3

1

d2d3

2

¹p|d3

1

d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

d3d1

p

1

4

¸d4D0D1D2D3D4D5

1

D1D2D4D5

2

¹tu,vu

Du

Dv

Φpu,vq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3 1 mod 4, D2D5, D1D4 1 mod 4. This is the exact set-up needed to apply

Theorem 1.6.

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| 1

2

¸d4

±Ei

1

E2E3

2

¹i,j

EiEj

Ψpi,jq,

139

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1 1 mod 4 and E2E3 1 mod 4.

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq2

We can multiply these together to get an expression for |HomσpGalpKurKq, Hj18 Cj2

4 Cj3

2 q|.We must get this back into the form in Theorem 1.6. As in Section 2.2.2 set Vj1,j2

t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand |HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q| as follows:

2pωpdq1qj32j1j2¸

d±Du

¸Ct1,...,j1j2u

χCj1,j2ppDuquPV q¹u,v

Du

Dv

Φj1,j2pu,vq

,

where the character is given by

χC,J1,J2

j1,j2ppDuquPV q

¹`PC,`¤j1

±u`1,2,4,5Du

2

¹`PC,`¡j1

±u`2,3Du

2

,

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

140

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du

$''&''%1 1 ¤ ` ¤ j1, i 0, 3

1 mod 4 1 ¤ ` ¤ j1, i 1, 2, 4, 5

¹u`i,i1

Du

$''&''%1 1 ¤ ` ¤ j1, i 0, 1

1 mod 4 j1 1 ¤ ` ¤ j1 j2, i 2, 3

For each partition k j1 j2 j3 and set C we can now apply Theorem 1.6 with M 8

and T ppZ8ZqqV describing the above congruence conditions to compute¸d X,d4 mod 8

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

The function Φj1j2 Ψj1j2 is indentical to the one found in Section 2.2.2 for the

case d 0 and d 1 mod 4. Thus the unlinked sets of maximal size have already been

classified to be of size 3j12j2 by Proposition 2.26. Therefore Theorem 1.6 implies the

main term only comes from j1 k, j2 j3 0, in which case Proposition 2.24 shows

that the maximal unlinked sets are exactly sets of the form U tu : ui P siu for a type

s P S tA,Buk. Therefore we can take j2, j3 0.

Thus Theorem 1.6 tells us that¸d X,d4 mod 8

|HomσpGalpKurKq, Hk8 q|

1

4k

¸Ct1,...,j1j2u

¸UγCpUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

2

3k22k

¸U

¸Ct1,...,j1j2u

¸UγCpUq

¸

d X,d4 mod 8

3kωpdq

OXplogXq3k2ε

141

The only remaining step is to compute the leading constant, where

γCpUq 43k¸

phuqPT pUqχCk,0pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies

¹u`i

tu

$''&''%1 mod 4 i 0, 3

1 mod 4 i 1, 2, 4, 5

As in Section 2.2.2, define the matrix Mk recursively. Then the y P F3k

2 satisfying the

above condition (written additively) for an independent set U of type s P S are solutions

of

¸u:uij

yu

$''&''%1 j 0, 3 and j P si, s`

0 else

By construction

Mky ¸

u:uijyu

pi,jq

which makes these conditions equivalent to Mky w for w P F3k2 . This set of solutions

is the coset y kerMk.

Lemma A.5. For all k ¥ 1 ¸UγC pUq 2k1.

Proof. Recall

χCk,0pphuquPV q ¹

`PC,`¤j1

±u`1,2,4,5 hu

2

k¹`1

p1qχCp`q°

u`1,2,4,5h2u1

8

The conditions on T are given modulo 4, so we can take a sum over all equivalence

classes of phuq P ppZ8Zqq|V | mod 4. Notice that

p5xq2 1

8 25x2 1

8 x2 x2 1

8 1 x2 1

8mod 2

142

for any odd x. Therefore relating hu12

xu P F2 we find that

¸pxuq mod 4

χCk,0ppxuquPV q ¸GU

k¹`1

p1qχCp`q°

u`1,2,4,5 χGpuq

¹uPUs

1 p1qχCp`qχt1,2,4,5upu`q

$''&''%23k C H

0 otherwise

Therefore we can write

¸C

¸UγCpUq 23k

¸sPS

¸xPykerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

This is equal to the°γCpUq found in the d ¡ 0, d 1 mod 4 case. Lemma A.4 shows

that this is 2k1, concluding the proof.

In conclusion, plugging in the value of this constant yields the result from Theorem

2.23:

¸d X,d4 mod 8

|SurjσpGalpKurKq, Hk8 q|

1

24k

¸d X,d4 mod 8

3kωpdq OXplogXq3k2ε

A.2.3 The case d ¡ 0, d 0 mod 8

If we include degenerate factorizations in Lemmermeyer’s classification of unramified H8-

extensions and apply Lemma A.3, then it follows for a quadratic field K of discriminant

d

|HomσpGalpKurKq, H8q| ¸

d8d1d2d3

81

2ap8d1, d2, d3q,

where the sum is over factorizations into odd, squarefree integers with 8d1, d2, d3 quadratic

discriminants (Note the factor of 8 is to account for automorphisms of H8 which lift to

143

G1. |HomσpGalpKurKq, H8q| is expanded as follows:

1

4

¸d8d1d2d3

1

d2d3

2

¹p|d3

1

2d1d2

p

¹p|d1

1

d2d3

p

¹p|d2

1

2d3d1

p

1

4

¸d8D0D1D2D3D4D5

1

D1D2D4D5

2

2

D2D4

¹tu,vu

Du

Dv

Φpu,vq

Here, Di are indexed by i P t0, ..., 5u and Φ : t0, ..., 5u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 3u. Moreover, these satisfy the congruence conditions

D0D3 1 mod 4, D2D5, D1D4 1 mod 4. This is the exact set-up needed to apply

Theorem 1.6.

In order to go from here to |SurjσpGalpKurKq, Hk8 q|, we need two pieces of informa-

tion. First, we can do an inclusion exclusion to show that

|SurjσpGalpKurKq, Hk8 q|

¸H¤Hk

8

µHk8pHq|HomσpGalpKurKq, Hq|,

where µHk8

is the Mobius function on the subgroup lattice as in Hall [29]. Second,

whenever µHk8pHq 0 (which implies H contains the Frattini subgroup of Hk

8 ), then H

is of the form

Hj18 Cj2

4 Cj32

for j1 j2 j3 k. This follows from the classification of subgroups of H8.

Fouvry-Kluners [25] showed in their work that

|HomσpGalpKurKq, C4q| 1

2

¸d8

±Ei

1

E2E3

2

2

E2

¹i,j

EiEj

Ψpi,jq,

where Ei are indexed by t0, 1, 2, 3uand Ψ : t0, 1, 2, 3u2 Ñ F2 is the characteristic function

of the set tpi, jq : 2 | j, i j, j 1u. Moreover, these satisfy the congruence conditions

E0E1 1 mod 4 and E2E3 1 mod 4.

144

Genus theory implies

|HomσpGalpKurKq, C2q| 2ωpdq2

We can multiply these together to get an expression for |HomσpGalpKurKq, Hj18 Cj2

4 Cj3

2 q|.We must get this back into the form in Theorem 1.6. As in Section 2.2.2 set Vj1,j2

t0, 1, 2, 3, 4, 5uj1 t0, 1, 2, 3uj2 , and define

Du gcdpDui , Euj1j: 1 ¤ i ¤ j1, 1 ¤ j ¤ j2q

In particular, it follows that Dp`qu ±

u`uDu for 1 ¤ ` ¤ j1, and E`u

±u`uDu for

j1 1 ¤ ` ¤ j2. This lets us expand |HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q| as follows:

2pωpdq1qj32j1j2¸

d±Du

¸Ct1,...,j1j2u

χCj1,j2ppDuquPV q¹u,v

Du

Dv

Φj1,j2pu,vq

,

where the character is given by

χCj1,j2ppDuquPV q ¹

`PC,`¤j1

±u`1,2,4,5Du

2

¹`¤j1

2±

u`2,4Du

¹

`PC,`¡j1

±u`2,3Du

2

¹`¡j1

2±

u`2Du

,

the function Φj1,j2 : V V Ñ F2 is given by

Φj1,j2pu,vq j1

`1

Φpu`, v`q j1j2¸`j11

Ψpu`, v`q,

and the tuples pDuq satisfy the congruence conditions

¹u`i,i3

Du

$''&''%±

uji,i3Du 1 ¤ `, j ¤ j1, i 0, 3

1 mod 4 1 ¤ ` ¤ j1, i 1, 2, 4, 5

¹u`i,i1

Du

$''&''%±

uji,i1Du 1 ¤ `, j ¤ j1, i 0, 1

1 mod 4 j1 1 ¤ ` ¤ j1 j2, i 2, 3

145

For each partition k j1 j2 j3 and set C we can now apply Theorem 1.6 with M 8

and T ppZ8ZqqV describing the above congruence conditions to compute¸d X,d0 mod 8

|HomσpGalpKurKq, Hj18 Cj2

4 Cj32 q|

The function Φj1j2 Ψj1j2 is indentical to the one found in Section 2.2.2 for the

case d 0 and d 1 mod 4. Thus the unlinked sets of maximal size have already been

classified to be of size 3j12j2 by Proposition 2.26. Therefore Theorem 1.6 implies the

main term only comes from j1 k, j2 j3 0, in which case Proposition 2.24 shows

that the maximal unlinked sets are exactly sets of the form U tu : ui P siu for a type

s P S tA,Buk. Therefore we can take j2, j3 0.

Thus Theorem 1.6 tells us that¸d X,d0 mod 8

|HomσpGalpKurKq, Hk8 q|

1

4k

¸Ct1,...,j1j2u

¸UγCpUq

¸odd n X

µ2pnq3kωpnq

OXplogXq3k2ε

1

3k22k

¸U

¸Ct1,...,j1j2u

¸UγCpUq

¸

d X,d0 mod 8

3kωpdq

OXplogXq3k2ε

The only remaining step is to compute the leading constant, where

γCpUq 43k¸

phuqPT pUqχCk,0pphuquPV q

¹tu,vu

p1qΦk,0pu,vqhu12

hv12 ,

such that t P T satisfies

¹u`i

tu

$''&''%±

uji tu mod 4 i 0, 3

1 mod 4 i 1, 2, 4, 5

146

As in Section 2.2.2, define the matrix Mk recursively. Then the y P F3k

2 satisfying the

above condition (written additively) for an independent set U of type s P S are solutions

of

¸u:uij

yu

$''&''%°

u:u`j yu j 0, 3 and j P si, s`

0 else

By construction

Mky ¸

u:uijyu

pi,jq

which makes these conditions equivalent to Mky w for 2 different w P F3k2 (One for

each possible value of°u`0,3 yu). This set of solutions is the union of the two cosets

y kerMk.

Lemma A.6. For all k ¥ 1 ¸UγC pUq 2k.

Proof. Recall

χCk,0pphuquPV q ¹

`PC,`¤j1

±u`1,2,4,5 hu

2

2±

u`2,4 hu

k¹`1

p1qχCp`q°

u`1,2,4,5h2u1

8°u`2,4

h2u1

8

The conditions on T are given modulo 4, so we can take a sum over all equivalence class

of phuq P ppZ8Zqq|V | mod 4. Notice that

p5xq2 1

8 25x2 1

8 x2 x2 1

8 1 x2 1

8mod 2

147

for any odd x. Therefore relating hu12

xu P F2 we find that

¸pxuq mod 4

χCk,0ppxuquPV q ¸GU

k¹`1

p1qχCp`q°

u`1,2,4,5 χGpuq°

u`2,4 χGpuq

¹uPUs

1 p1qχCp`qχt1,2,4,5upu`qχt2,4upu`q

$''&''%23k C ti : si Bu

0 otherwise

Therefore we can write

¸C

¸UγCpUq 23k

¸sPS

¸y

¸xPykerMk

¹tu,vu

p1q°

siB Φpui,viqxuxv

This is equal to the sum over the 2 choices of coset y kerMk of the°γCpUq found in

the d ¡ 0, d 1 mod 4 case. Lemma A.4 shows that this is 2k1, so multiplying by 2

concludes the proof.

In conclusion, plugging in the value of this constant yields the result from Theorem

2.23:

¸d X,d0 mod 8

|SurjσpGalpKurKq, Hk8 q|

1

24k

¸d X,d0 mod 8

3kωpdq OXplogXq3k2ε

148

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