Periods of Eisenstein series and cohomology of modularcurves

Ed Belk

December 29, 2016

2

Contents

0 Introduction 9

1 Preliminaries 11

1.1 The upper half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Modular forms and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Periods and cycle-cocycle pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 The approach of Glenn Stevens 21

2.1 Periods: a new definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Periods and special values of L-functions . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 An ancillary basis of Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Fourier coefficients of Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Periods of basis elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 The adelic picture 35

3.1 The classical-adelic correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Integration on SL2.R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Integration on adelic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A Lattices and elliptic curves 51

B Periods 53

C The period of a cusp form 55

D Supplementary calculations 59

3

4

To my mother

5

6

Acknowledgements

The author gratefully recognizes the tremendous assistance offered by the faculty of the mathe-matics department at the University of British Columbia, particularly that of Julia Gordon, GregMartin, Lior Silberman, and Vinayak (Nike) Vatsal, without whose guidance this thesis would nothave been possible.

Furthermore, the encouragement offered by the faculty of the department of mathematics andstatistics at Queen’s University was essential to the mathematical base on which this work ispredicated. Personal thanks are extended to Ivan Dimitrov, Abdol-Reza Mansouri, Ram Murty,Mike Roth, Greg Smith, and Noriko Yui for their support which far exceeded that which wasrequired or requested.

Finally, the author wishes to thank Jack Weiner of the University of Guelph for taking the time toaddress the calculus problems of a fourteen-year-old.

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8

Chapter 0

Introduction

Among the maxims on LordNaoshige’s wall was this one: ‘Mattersof great concern should be takenlightly.’ Master Ittei commented:‘Matters of small concern should betaken seriously.’

—Yamamoto Tsunetomo, Hagakure

The goal of this thesis is to define the period of an Eisenstein series by transferring the definitiongiven by Glenn Stevens to the modern, adelic setting.

The space of modular forms (of a given weight and level, say) decomposes naturally as the directsum of the space of cusp forms, and the space of Eisenstein series (section 1.2). By definition, thecusp forms are those modular forms which vanish at the cusps Q[ fi1g of H; this, combined withthe requirement of rapid decay as y !1, means that cusp forms are integrable on contours in theupper half-plane H joining any pair ˛ and ˇ, where ˛; ˇ 2 Q [ fi1g (section 1.3).

It may be shown (appendix C) that a rational number may be written as a sum of Farey neighbours,and moreover that if ˛; ˇ 2 Q are Farey neighbours, then the geodesic in H joining ˛ and ˇ is G.Z/-equivalent to the contour joining 0 to i1; thus it suffices to investigate integration on this contour.The period of a cusp form f .z/ dz is defined (section 1.3, also appendix C) to beZ i1

0

f .z/dz: (1)

Of course, if f .z/ is an Eisenstein series, this integral will not converge: to see this it is enough toobserve that the constant term in the Fourier expansion of f .z/ is nonzero.

However, as demonstrated by Glenn Stevens in [Ste12], the periods of cusp forms can be expressedin terms of various well-defined analytic and meromorphic functions associated to modular formsin general (sections 2.1–2.2), and so we are led to investigate the values of these expressions whenf .z/ is an arbitrary modular form. Ultimately, we define the period of an Eisenstein series usingsuch relations, and use them to compute the values of periods of Eisenstein series (sections 2.3–2.6).

9

The explicit correspondences between quotients of H by the action of arithmetic subgroups � �SL2.Z/, and the quotients of adelic algebraic groups, allow us to investigate in the adelic settingthe effect of Stevens’ regularization procedure (section 3.1), and to understand the constructionfrom this perspective.

By relating measures on H to measures on the adelic quotient (section 3.2), we are able (section 3.3)to express the value of the period of an arbitrary (classical) modular form solely in terms of adelicintegrals. Finally (section 3.4), we briefly review the modern theory of automorphic forms, and seehow it is used to realize classical modular forms. As such, we may then reverse our construction,and define the period of an (automorphic) Eisenstein series, using Stevens’s method, but withoutreference to the classical theory.

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Chapter 1

Preliminaries

In order to state our problem, we begin by reviewing some of the theory of modular forms, modularfunctions, and the upper half-plane, as well as some basic facts about modular curves, closing insection 1.3 with an explanation of the main goal of this thesis. The primary reference is [Mil90],��I.2–4.

1.1 The upper half-plane

The upper half-plane H WD fz D x C iy 2 C W y > 0g is the initial setting for our investigation.It is equipped with the Poincare metric

ds2 D1

y2dz d Nz D

.dx/2 C .dy/2

y2

to form the Poincare upper half-plane model. Together with an atlas with the single chart H,this gives a Riemann surface which is conformally equivalent to the open unit disk; we will discussthe analytic properties of H in section 1.3.

We consider throughout the algebraic group G D SL2 and its Borel subgroup B consisting of theupper triangular matrices; we caution that the notation B is also used below for the subgroupof upper-triangular matrices with positive diagonal entries. The group G.R/ acts on H by linearfractional transformations: �

a b

c d

�� z D

az C b

cz C d;

which naturally induces actions of the subgroups G.Z/; G.Q/, etc. We observe that�1 x

1

��y1=2

y�1=2

�� i D

�y1=2 xy�1=2

y�1=2

�� i D x C iy; (1.1)

and so the action of G.R/ is transitive on H. As the stabilizer of i 2 H under this action isK1 WD SO2.R/, we obtain a one-to-one correspondence G.R/=K1 Š H:�

y1=2 xy�1=2

y�1=2

�K1 2 G.R/=K1

� ! x C iy 2 HI

11

we will exploit this correspondence below. The notation K1 is motivated by the adelic approach(section 3.1) in which we deal with subgroups Kv for every valuation v of Q.

The action of G.Z/ on the upper-half plane (i.e. the restricted action of G.R/) naturally extends tothe boundary P1.Q/ D Q[fi1g by the same formula; we call this set the cusps of H. If � � G.R/is an arbitrary discrete subgroup, then we might reasonably ask which cusps are inequivalent underthe action of �, i.e. which cusps descend to unique orbits in the quotient �nH.

We also define the principal congruence subgroups for N � 1:

�.N/ D f 2 G.Z/ W � I mod N g;

the congruence taken componentwise (so that, for instance, �.1/ D G.Z/). By a congruencesubgroup we mean a subgroup � of G.Z/ such that �.N/ � � for some N � 1; if N is the leastsuch integer for which this is the case, � is said to be a congruence subgroup of level N .

We note that the principal congruence subgroups have finite index in G.Z/: the Chinese re-mainder theorem implies ([Cas12], theorem 1.1) that the canonical componentwise projectionG.Z/ ! G.Z=NZ/ is surjective for every N � 1, and because �.N/ is precisely the kernel ofthis map, one has by the first isomorphism theorem Œ�.1/ W �.N/� D jG.Z=NZ/j, which is evidentlyfinite.

We also remark here that, because the action of the centre Z�G.R/

�D f˙I g (where I 2 G.R/

is the identity) on H is trivial, the action of G.R/ on H is more naturally thought of as being anaction of G.R/=f˙I g Š PGL2.R/. Indeed, we must consider the action of this group if we wishto find a fundamental domain for PGL2.Z/ on which to work, as the action of G.Z/ itself is notfaithful, while that of PGL2.Z/ is. Where there will be no confusion, we will identify elements ofPGL2.R/ with elements of G.R/ representing them.

We claim that a fundamental domain for the action of PGL2.Z/ on H is given by the set

D D fz 2 H W jzj > 1;�12< Re.z/ < 1

2g:

Indeed, given z 2 H, we must have Im.z/ � 1 or Im.z/ < 1. Thus if

S D

��1

1

�; T D

�1 1

1

�; so S � z D

�1

zand T n � z D z C n;

and m 2 Z is the unique integer such that m � 12� Re.z/ < mC 1

2, then we must have

T �m � z 2 D or ST �m � z 2 D;

where D is the closure of D. It may be shown ([Mil90], pp. 32–33) that PGL2.Z/ has presentationhS; T jS2 D .ST /3 D 1i, and more precisely which elements of @D are �.1/-equivalent; however, wewill not need D extensively, and so we will not pursue this here.

We see that D is given by an “ideal triangle” with vertices at �; �2; (where � D 1Cip3

2), and i1.

In practice, we will be more concerned with a different ideal triangle D, with vertices at 0, 1, andi1; while not a fundamental domain itself, the closure of D is the closure of the union of threetranslates of a particular fundamental domain. Indeed, let D0 � D be the subset consisting of thosez 2 D with 0 < Re.z/ < 1

2; jzj < 1g: then

D D D0 [ .ST �1/ �D0 [ .ST �1/2 �D0:

12

As such, the G.Z/-translates of D cover H; this is known as the Farey tessellation of H. Thetranslates of @D are geodesics in H of two sorts: the first are the vertical contours fnC i t W t > 0g,where n 2 N, and the second are semicircles joining Farey neighbours, i.e., rational numbersab> p

qsuch that aq � bp D 1; the Farey tessellation will prove very useful in simplifying our

problem (appendix C). We will denote by f˛; ˇg the geodesic joining ˛ to ˇ; as H is simply-connected, this serves as a representative for the homotopy class of contours in H (or H�) joining˛ and ˇ.

Finally we introduce two modular curves associated to �, namely the affine curve Y.�/ WD �nHand its compactification, the projective curve X.�/ WD �nH�. These are of significant interestin modern number theory due to the fact that they parameterize isomorphism classes of ellipticcurves over Q (see [Mil90], section 8; also appendix A below). While we will not explore the deeperimplications of this fact, we will use the modular curves to contextualize our main problem, whichwe introduce in the following sections.

1.2 Modular forms and functions

The upper half-plane arises naturally in the context of differential geometry: it is an immediatecorollary of the uniformization theorem ([Poi08], pp. 21-50) that H is the universal covering spacefor any Riemann surface of constant negative curvature. Thus to study holomorphic (or meromor-phic) functions on such Riemann surfaces, it suffices to investigate such functions on H.

Let D

�a b

c d

�2 �, where � � G.Z/ is some subgroup of finite index; we might ask which

differentials f .z/ dz on H are invariant under the action of , and which therefore descend tomeromorphic differentials on �nH. We may compute

f . � z/d. � z/ D f . � z/.cz C d/�2 dz DW j. ; z/�2f . � z/dzI

we call j. ; z/ D .cz C d/ the factor of automorphy, and record the identities

j.g1g2; z/ D j.g1; g2 � z/j.g2; z/ andd

dz. � z/ D j. ; z/�2:

We also introduce ([New14], p. 13) the slash operator (of weight 2) on the space of functions onH: given ˛ D

�a bc d

�2 G.Q/, we define

.f j˛/.z/ WD j.˛; z/�2f .˛ � z/:

The slash operator arises naturally in the context of contour integrals on H: we observe thatZ ˛�z1

˛�z0

f .z/dz D

Z z1

z0

f .˛ � z/d.˛ � z/ D

Z z1

z0

f .˛ � z/j.˛; z/�2 dz D

Z z1

z0

.f j˛/.z/dz; (1.2)

and that f j.˛ˇ/ D .f j˛/jˇ. It may be shown (appendix D) that the slash operator induces anaction of G.Q/ on the space of modular forms for congruence subgroups of �.1/.

It is apparent that the only functions f .z/ on H for which the differential f .z/ dz is invariant underthe action of � are those which satisfy f . � z/ D j. ; z/2f .z/ (or, equivalently, f j D f ) for all 2 �.

13

Now, let us recall the notion of a modular function for an arithmetic subgroup � � G.Z/: thisis a function f W H! C satisfying

1. f is meromorphic on H;

2. f . � z/ D f .z/ for all z 2 H; 2 �;

3. f is meromorphic at infinity.

This last condition is understood to mean that the function f .�1=z/ has, at worst, a pole at zero.

We also recall the notion of a modular form of weight 2k (k � 0) for � � G.Z/. This is afunction f W H! C satisfying

1. f is holomorphic on H;

2. f . � z/ D j. ; z/2kf .z/ for all z 2 H; 2 �;

3. f is holomorphic at infinity.

Any function satisfying condition 2. is called weakly modular of weight 2k. A modular formwhich satisfies f .i1/ WD lim

y!1f .z/ D 0 is known as a cusp form, so called because it extends

uniquely to a function on H� which vanishes at the cusps.

Evidently, the quotient of two nonzero modular forms of weight 2k for � is a modular function for�; we will encounter examples of such functions below. If N is the least number such that f is amodular form for �.N/, then f is said to have level N .

If, in conditions 1. and 3. of our definition of a modular form, we replace the word “holomorphic”with “meromorphic,” we obtain the so-called meromorphic modular forms; in this document,all so-called “modular forms” will be assumed holomorphic unless otherwise stated. We remarkthat if f is a (holomorphic or meromorphic) modular form of weight 2 for � 3 ˛, then f j˛ D f .

Now, suppose f is weakly modular of level N , of any weight. As�1 N1

�2 �.N/ � � and

�1 N1

��z D

z C N , we see that condition 2. implies that f must be periodic with period N , and so we maytake its Fourier series:

f .z/ D

1XnDm

anqn=N ; q D e2�iz; m 2 Z [ f�1g:

The condition meromorphic at infinity means, in this situation, that m is finite, and holomorphicat infinity that m � 0.

The vector space of all modular forms of weight 2k for � is denoted M2k.�/, and the space of cuspforms of weight 2k for � is similarly denoted S2k.�/. Taking the direct limit over the direct systemof congruence subgroups of finite index ordered by inclusion gives us the algebra of modular forms(respectively, cusp forms) of weight 2k for all levels:

M2k WD lim�!

M2k

��.N/

�and S2k WD lim

�!S2k

��.N/

�:

14

We note that the linear map M2.�/ ! C defined by f 7! f .i1/ has kernel S2.�/; below, wewill construct modular forms G2k.z/ which do not vanish at i1, thus implying that this map issurjective, and that therefore

M2.�/ D S2.�/˚ E2.�/

for some subspace E2.�/. We call this subspace the space of Eisenstein series of weight 2 for �.

For a concrete example of a modular form of weight 2k for �.1/, k � 2, we take the Eisensteinseries of weight 2k: in this context, this is the series defined by

G2k.z/ DX

.m;n/2Z2

0 1

.mz C n/2k;

the prime over summation indicating here and everywhere below that the possibility m D n D 0 isexcluded. This series converges absolutely for z 2 H� when k � 2, and satisfies for every 2 �.1/the relation

G2k. � z/ D j. ; z/2kG2k.z/:

As for holomorphy, we cite the following

Proposition 1.2.1 ([Mil90], p. 57). Let k � 2; z 2 H, and q D e2�iz. The Eisensteinseries for �.1/ has Fourier expansion

G2k.z/ D 2�.2k/

�1C

2

�.1 � 2k/

1XnD1

�2k�1.n/qn

�;

where �k.n/ DPd jn d

k is the generalized sum-of-divisors function and �.s/ is theRiemann zeta function, and this series converges absolutely.

It therefore follows that each G2k; k � 2, is a modular form, which is not a cusp form (as G2k.i1/ D2�.2k/). In the remainder of this section, we will investigate the properties of G2k.z/ and similarfunctions to familiarize ourselves with some of the techniques of working with Eisenstein series.

Remark: While our calculations are only valid for weight at least 4, they nonetheless illuminateby analogy the calculations we perform in the next chapter on Eisenstein series of weight 2.

One sometimes encounters Eisenstein series defined by summing over different indexing sets; forinstance, by the formula

E2k.z/ D1

2

X.c;d/D1

1

.cz C d/2k;

the summation taken only over relatively prime integers c; d . As we will see in later sections, thisindexing set is in some ways more natural than that of G2k.z/: pairs of relatively prime integers c; dexist in one-to-one correspondence with certain cosets of �.1/, allowing us to define the Eisensteinseries without reference to the upper half-plane.

We resolve the apparent discrepancy in the the two definitions with a

Proposition 1.2.2. Let G2k.z/ and E2k.z/ be the Eisenstein series for �.1/ definedabove; then

G2k.z/ D 2�.2/E2k.z/ D 2�.2/X

�1 2�1n�.1/

j. ; z/�2k;

15

where �.s/ is the Riemann zeta function and �1 � �.1/ is the subgroup of upper-triangular matrices.

Proof. By absolute convergence for k � 2, we have

G2k.z/ DX

.m;n/2Z2

0 1

.mz C n/2kD

X.m;n/2Z2

.m;n/D1

1

.mz C n/2kC

X.m;n/2Z2

.m;n/>1

1

.mz C n/2k

D 2E2k.z/C

1XdD2

X.m;n/2Z2

.m;n/Dd

1

.mz C n/2kD 2E2k.z/C

1XdD2

X.m0;n0/2Z2

.m0;n0/D1

1

.m0dz C n0d/2k

D 2E2k.z/C

� 1XdD2

1

d2k

�� X.m0;n0/2Z2

.m0;n0/D1

1

.m0z C n0/2k

�

D 2E2k.z/C 2��.2k/ � 1

�E2k.z/;

thus

G2k.z/ D 2�.2/E2k.z/; and so E2k.z/ D 1C2

�.1 � 2k/

1XnD1

�2k�1.n/qn:

This proves the first equality. For the second, it suffices to prove that relatively prime pairs ofintegers exist in one-to-one correspondence with the cosets �1n�.1/. To this end, suppose c; d 2 Zwith .c; d/ D 1; we know by Bezout’s identity that there exist integers x; y such that cxC dy D 1,and therefore that �

y �x

c d

�2 �.1/:

Moreover, all solutions to cx0 C dy0 D 1 are given by x0 D x � td; y0 D y C tc with t 2 Z, andtherefore all matrices in �.1/ with bottom row .c d/ are of the form�

1 t

1

��y �x

c d

�D

�y C tc �.x � td /

c d

�;

which proves our claim. It now follows immediately that

G2k.z/ D 2�.2/X

2�1n�.1/

j. ; z/�2k;

where we have abused notation by writing D �1 2 �1n�.1/.

For an arbitrary subgroup � � �.1/ of finite index, we define the space of Eisenstein series ofweight 2k, denoted E2k.�/, to be the orthogonal complement of the space of cusp forms under theevaluation-at-i1 map; by definition, S2k.�/ is the kernel of this map (appendix C).

The Eisenstein series are the modular forms which will be of greatest interest to us. By dimensionalconsiderations ([SG07], chap. 6) it may be shown that the Eisenstein series we have just constructedgive all the Eisenstein series E2k

��.1/

�; k � 2, and in the next chapter we will in a similar way

construct generating sets for all E2��.N/

�; N � 2; it is a standard result that E2

��.1/

�D 0.

16

We close by constructing (one version of) the Eisenstein series of weight 2k and level N when k � 2.Following [Bro11], fix k and let

E2k.zIN/ DX

2�0.N/n�.N/

j. ; z/�2k; (1.3)

where �0.N / D �1 \ �.N/. This may be thought of as the Eisenstein series for �.N/ associatedto the cusp i1; if �1 D i1; �2; : : : ; �n are the inequivalent cusps of �.N/nH�, then we may fix�i 2 G.R/ so that �i � i1D �i , and define the Eisenstein series associated to the cusp �i via

E.i/

2k.zIN/ D

X 2�0.N/n�.N/

j.��1i ; z/�2k :

The definition of E.i/

2k.zIN/ has the advantage of familiarity, due to its similarity with the definition

of E2k.z/ above. However, as we will see in the next chapter, their analogues Ga.z/ in the weight2 case are unsuitable for our purposes, but are still useful indirectly for constructing the basis �xof the space of weight 2 Eisenstein series which we will ultimately use.

Proposition 1.2.3. Let E2k.zIN/ D E.1/

2k.zIN/ be the Eisenstein series defined in

equation (1.3); then

E2k.zIN/ DX

.c;d/D1c�0 mod Nd�1 mod N

1

.cz C d/2k:

Proof. It suffices to show that the elements of �0.N /n�.N/ are in one-to-one correspondence withthe set

fi1g [ f cd2 Q W .c; d/ D 1; d � 1 mod N; c � 0 mod N g:

The calculation resembles our earlier one: every element

�y �x

c d

�of �.N/ gives a solution to the

equationcx C dy D 1;

with d � y � 1 mod N and c � x � 0 mod N , and taking this equation modulo N yieldsy � 1 mod N . The formulas

x0 D x C dt; y0 D y � ct

imply thatx0 � x C t mod N and y0 � y mod N;

and so only for t a multiple of N does .x0; y0/ have

cx C dy D 1; d � y � 1 mod N; c � x � 0 mod N:

That is, all matrices in �.N/ with bottom row .c d/ have the form�1 tN

1

��y �x

c d

�D

�y C tcN �.x � tdN /

c d

�for some t 2 Z.

17

This proves one inclusion; for the converse, let .c; d/ D 1 with c � 0 mod N; d � 1 mod N . ByBezout’s identity there exist integers x; y so that cx C dy D 1, and our work above shows thaty � 1 mod N and that x may be shifted by an integer multiple of d so that x0 D x � td is zeromodulo N . That is, there exists a matrix in �.N/ with bottom row .c d/, and we are done.

We have now seen how Eisenstein series of weight at least 4 may be defined independently of theupper-half plane; we will return to these considerations in chapter 3, when we will attempt thesame for weight 2.

1.3 Periods and cycle-cocycle pairings

We observe that modular functions and modular forms, while defined as functions on H, are natu-rally identified as functions and differentials, respectively, on the modular surfaces X.�/ and Y.�/:this is an immediate consequence of the �-invariance in the former case, and the �-invariance ofthe differential f .z/ dz in the latter case. Indeed, it may be shown ([Mil90], p.51) that there is aone-to-one correspondence between:

1. the set of meromorphic modular forms for � on H�;

2. the set of meromorphic differential forms on X.�/; and

3. the set of meromorphic, �-invariant differential forms on H�.

In particular, the modular forms for � on H� descend to holomorphic differential forms on theprojective, algebraic curve X.�/ D �nH�. These forms $ D f .z/dz; f .z/ 2 S2.�/ representcohomology classes in the first de Rham cohomology group H 1

�X.�/IC

�WD H 1

dR

�X.�/IC

�, whose

elements we identify with the closed, holomorphic 1-forms on X.�/ modulo the exact 1-forms.

Dually we have the first homology group H1�X.�/IC

�, whose elements are equivalence classes

of closed, continuous paths in X.�/ modulo homotopy. Denoting elements of the former group byrepresentatives f .z/dz and elements of the latter by path representatives Œz0; z1�, we have a naturalpairing

H1�X.�/IC

��H 1

�X.�/IC

�C�

Œz0; z1�; f .z/dz� Z z1

z0

f .z/dz

Each of these integrals converges when f .z/ is a cusp form, but not if f .z/ is an arbitrary modularform. The analogous pairing for Y.�/ is only valid when the (preimages of the) endpoints of ourcontour lie in H, and not when they lie on its boundary. Thus we face the problem of attemptingto resolve this obstruction.

In particular, our difficulty arises when we attempt to integrate Eisenstein series over contours inX.�/ which pass through the cusps; evidently it suffices to consider contours whose endpoints arecusps of X.�/. The values of integrals between cusps are determined by the values of any oneparticular such integral (appendix C), and so it suffices to consider the single integralZ i1

0

E.z/dz; (1.4)

18

when E.z/ is an Eisenstein series. If in equation (1.4) we replace E.z/ with a cusp form f .z/, thenthe integral is called the period of the cusp form f .z/, for reasons which are explained in appendixC.

We are therefore confronted with the problem of defining the period of an Eisenstein series. Tothis end, we investigate the work of Glenn Stevens, who uses the theory of group cohomology toobtain an effective definition of the period of an Eisenstein series, which moreover allows one toeasily compute these periods for various bases of these spaces.

19

20

Chapter 2

The approach of Glenn Stevens

In this chapter we follow the work of Glenn Stevens ([Ste12], chapter 2) to define a certain function�f on G.Q/ whose value, when f is a cusp form, on particular elements of G.Q/ coincides withthe value of the period of f , as defined in appendix C; this is accomplished in section 2.1.

We then concern ourselves with computing the value of these periods: in section 2.2 we express thevalues of our newly-defined periods in terms of special values of L-functions and constant termsof Fourier expansions. In section 2.3 we reduce the scope of our considerations to a particular(countable) set of Eisenstein series, which we construct in section 2.4. This set will not be usefulfor directly computing our periods, but it will allow us to construct another set which will allow usto do so.

In section 2.5 we obtain basis elements for our subspace of Eisenstein series and compute theirFourier coefficients. Finally, in section 2.6, we combine our work from the preceding sections tostate our main results (theorem 2.6.1 and its corollary), by which we achieve our goal of finding aneffective definition of the period of an Eisenstein series.

2.1 Periods: a new definition

We begin by recalling the slash operator from chapter 1; we note thatZ .˛ˇ/�z0

z0

f .z/ dz D

Z ˇ �z0

z0

.f j˛/.z/ dz C

Z ˛�z0

z0

f .z/dz; (2.1)

and, as we saw in equation (1.2), thatZ z1

z0

.f j˛/.z/dz D

Z ˛z1

˛z0

f .z/dz: (2.2)

We will use equation (2.2) extensively in our calculations below.

Let G D SL2 as before. The domain of the slash operator may be naturally extended to thegroup ring CŒG.Q/� by linearity ([Ste12], p. 48) giving an action of CŒG.Q/� on M2: we will writef j.a 1 C b 2/ D af j 1 C bf j 2 for a; b 2 C; 1; 2 2 G.Q/.

21

Now, fix some z0 2 H; for every f 2M2 we have a map 'f W G.Q/! C given by

'f .˛/ D

Z ˛�z0

z0

f .z/dz:

In particular, this means there is a map ' W G.Q/ ! HomC.M2;C/ DWM�2, where

�'.˛/

�.f / WD

'f .˛/.

The group G.Q/ (and thus CŒG.Q/� by extension) also acts (on the right) on the set of all such 'fby the rule ˛:'f D 'f j˛; we may now rewrite (2.1) as

'f .˛ˇ/ D 'f .˛/C .˛:'f /.ˇ/:

That is, 'f satisfies the 1-cocycle condition of G.Q/ in C for every f , and so ' is a crossedhomomorphism from G.Q/ to M�

2; it therefore represents a cohomology class in H 1�G.Q/IM�

2

�.

We recall from section 1.2 that a modular form f .z/ of level N has Fourier expansion

f .z/ D

1XnD0

anqn=N ;

where q D e2�iz; we introduce the notation a0.f / WD a0 and Qf .z/ D f .z/ � a0.f /.

Now, let ˛; f; z0 be as above. We consider the map �f W G.Q/! C given by

�f .˛/ WD

Z ˛z0

z0

f .z/dz � z0a0�f j.˛ � 1/

�C

Z i1

z0

Cf j.˛ � 1/.z/dz:

The definition of �f is independent of z0:

d

dz0

�Z ˛z0

z0

f .z/ dz � z0 � a0�f j.˛ � 1/

�C

Z i1

z0

Cf j.˛ � 1/.z/dz

�D f .˛ � z0/

d

dz0.˛ � z0/ � f .z0/ � a0

�f j.˛ � 1/

��Cf j.˛ � 1/.z0/

D f .˛ � z0/j.˛; z0/�2� f .1 � z0/j.1; z0/

�2� f j.˛ � 1/.z0/ D 0:

Furthermore, each �f satisfies the 1-cocycle condition: as

�f .˛ˇ/ D 'f .˛ˇ/ � z0a0�f j.˛ˇ � 1/

�C

Z i1

z0

Df j.˛ˇ � 1/.z/dz;

it suffices to prove the claim for each summand. This is already established for 'f .˛ˇ/, and forthe second term we have

a0�f j.˛ˇ � 1/

�D a0

�f j.˛ˇ � ˛ C ˛ � 1/

�D a0

�f j.˛ˇ � ˛/C f j.˛ � 1/

�D a0

�f j.˛.ˇ � 1//

�C a0

�f j.˛ � 1/

�D a0

�.f j˛/j.ˇ � 1/

�C a0

�f j.˛ � 1/

�D .˛:a0/

�f j.ˇ � 1/

�C a0

�f j.˛ � 1/

�;

22

as claimed. Finally for the last summand one hasZ i1

z0

Df j.˛ˇ � 1/.z/ dz D

Z i1

z0

ef j.˛ˇ � ˛ C ˛ � 1/.z/ dz

D

Z i1

z0

Bf j.˛ˇ � ˛/C f j.˛ � 1/.z/dz D

Z i1

z0

�Ef j�˛.ˇ � 1/�.z/CCf j.˛ � 1/.z/

�dz

D

Z i1

z0

E.f j˛/j.ˇ � 1/.z/dz C

Z i1

z0

Cf j.˛ � 1/.z/dz

D ˛:

�Z i1

z0

Cf j.ˇ � 1/.z/ dz

�C

Z i1

z0

Cf j.˛ � 1/.z/ dz;

and we are done. Thus � similarly represents a cohomology class in H 1�G.Q/IM�

2

�.

We observe that, if f 2M2.�/ and ˛ 2 �, then

�f .˛/ D

Z ˛z0

z0

f .z/ dz D 'f .˛/:

We might be tempted now to take a limit z0 ! 0 to obtain a definition of the integral of anarbitrary modular form from 0 to i1; however, this operation is not well-defined in the Poincareupper-half plane.

Nonetheless, if f is a cusp form, the expression defining �f .˛/ is well-defined even when z0 D 0,and moreover, for ˛ 2 G.Q/ such that ˛ � 0 D i1, is precisely the period integral of f , as definedin appendix C.

Therefore, because �f is well-defined for arbitrary f 2M2, we are thus motivated to define the

period of a modular form f as the value of �f .˛/, where ˛ � 0 D i1. Evidently, ! D

��1

1

�is

such an element, and taking z0 D i , we have for any cusp form f that

�f .!/ D 'f .!/ D

Z i

i

f .z/ dz � ia0�f j.! � 1/

�C

Z i1

i

Cf j.! � 1/.z/ dz

D

Z i1

i

f j.! � 1/.z/dz;

because f j.! � 1/ is again a cusp form. ThusZ i1

i

f j.! � 1/.z/dz D

Z i1

i

.f j!/.z/dz �

Z i1

i

f .z/dz

D

Z 0

i

f .z/dz �

Z i1

i

f .z/dz D �

Z i1

0

f .z/ dz;

which is (up to sign) precisely our period integral.

Definition. Let f 2M2 be a modular form. Then the period of f is defined to bethe value of �f .!/.

23

Having achieved our first goal of finding an effective definition for the period of a modular form,we move on to another goal: to compute all of them.

In closing, we remark that �f .!/ may be considered as a sort of principal value integral: we havedemonstrated the relationship of �f and 'f , and the independence of the definition of �f of choiceof z0. Taking z0 D i t to be purely imaginary, we have

'f .!/ D

Z !�it

it

f .z/dz D �

Z it

i=t

f .z/ dz:

By taking t > 1, we obtain a multiplicatively symmetric contour fi=t; i tg on which we integrate themodular form f .z/. Taking t arbitrarily large does not affect the value of �f .!/, and so taking thelimit allows us to express the period of f in terms of the principal value integral by the relationZ P

f0;i1g

f .z/ dz WD limt!1

��f .!/C i ta0

�f j.˛ � 1/

��

Z i1

it

Cf j.˛ � 1/.z/dz

�:

2.2 Periods and special values of L-functions

In this section we will realize the values of �f .!/ as special values of L-functions; this is establishedin our main result, theorem 2.2.5, which is the culmination of the preceding results. In order toexplicitly compute the values of �f , we must first introduce several ancillary functions and performa calculus of special values, which we begin now.

First of all: let f 2M2 be a modular form of weight 2 and level N . Recall from section 2.1 theconstant term function a0.f / and the “truncation” Qf .z/ D f .z/ � a0.f / of f .z/, and write

f .z/ D

1XnD0

anqn=N :

We observe that

a0.f / D1

N

Z N

0

f .z/ dz;

and we record a standard

Proposition 2.2.1 ([Ste12], lemma 2.1.1). Let f be as above, and let " > 0. Onehas

(a) an D O.n1C"/ as n!1;

(b) f .iy/ D O.y�2�"/ as y ! 0;

(c) Qf .iy/ D O.e�y=N / as y !1.

Proof. (a) [Hec37], Satz 6.

(b) [Ogg69], p. I-3

(c) [Ste12], p. 45

24

We define the L-function associated to f first to be the Dirichlet series

L.f I s/ D N s1XnD1

an

ns:

By part (a) of proposition 2.2.1, this series converges absolutely in Re.s/ > 2, and we will showthat it continues meromorphically to C with only possible simple poles at s D 0 and 2. To do this,we define first the Mellin transform of f :

D.f; s/ D

Z i1

0

Qf .z/ys�1 dz;

where y D y.z/ D Im.z/. Parts (b) and (c) of proposition 2.2.1 imply that D.f; s/ convergesabsolutely in Re.s/ > 2 and so defines an analytic function of s in this domain.

Proposition 2.2.2 ([Ste12], proposition 2.1.2). Let ! D��1

1

�2 �.1/. One has

(a) D.f; s/ D i�.s/.2�/�sL.f; s/.

(b) D.f; s/ D

Z i1

i

Qf .z/ys�1 dz �

Z i1

i

ef j!.z/y1�s dz C i�a0.f j!/

2 � s�a0.f /

s

�.

(c) D.f; s/ continues analytically to a meromorphic function of s with possible simplepoles only at s D 0 and s D 2, and satisfies the functional equation

D.f; s/CD.f j!; 2 � s/ D 0:

Proof. We follow [Ste12] (pp. 49–50) exactly:

(a) For Re.s/ > 2, we have

D.f; s/ D

Z i1

0

� 1XnD1

anqn=N

�ys�1 dz D i

1XnD1

an

Z 10

e�2�ny=Nysdy

y:

The change of variables y 7! 2�nyN

gives

D.f; s/ D i

1XnD1

an

�N

2�n

�s Z 10

e�yysdy

yD i�.s/

�N

2�

�s 1XnD1

an

ns:

(b) For Re.s/ > 2, we have

D.f; s/ D

Z i

0

Qf .z/ys�1 dz C

Z 1i

Qf .z/ys�1 dz:

Using equation (2.2), we computeZ i

0

Qf .z/ys�1 dz D

Z i

0

�f .z/ � a0.f /

�ys�1dz D

Z i

0

f .z/ys�1 dz � ia0.f /

s:

25

We have !�1 � 0 D i1; !�1 � i D i , thusZ i

0

f .z/ys�1 dz D

Z 0

i1

f .! � z/y.! � z/s�1 d.! � z/ � ia0.f /

s

D �

Z i1

i

.f j!/.z/y.! � z/s�1 dz � ia0.f /

s:

We observe that y.! � z/ D y��1z

�D

y.z/

jzj2, and so in particular on our contour (when z is

purely imaginary) we have y.! � z/ D y.z/�1, thusZ i

0

Qf .z/ys�1 dz D �

Z 1i

.ef j!/.z/y1�s dz C ia0.f j!/

2 � s� i

a0.f /

s;

and the result is proven.

(c) Both integrals in part (b) converge absolutely for all s.

The Mellin transform and the constant terms a0.f / allow us to express neatly the distinguishedvalues L.f; 0/ and L.f; 1/.

Proposition 2.2.3 ([Ste12], proposition 2.2.1). Let f be a modular form of weight2 and level N with associated L-function L.f; s/. Then

1. L.f; 0/ D �a0.f /.

2. L.f; 1/ D �2�i �D.f; 1/.

Proof. By proposition 2.2.2 (a), we know

D.f; s/ D i�.s/.2�/�sL.f; s/:

We now obtain the proposition by evaluating residues at s D 0 and 1.

Now: recall the action of G.Q/ on M2 and its extension by linearity to the action of the group ringCŒG.Q/�, retaining the notation ˛:f D f j˛. For f 2M2, define the function

ef W CŒG.R/� �! C

by the rule

ef .˛/ D D.f j˛; 1/:

The function ef will allow us to express our period integrals cleanly; we record its properties below.

Proposition 2.2.4 ([Ste12], proposition 2.2.2). Let ˛; ˇ 2 CŒG.Q/� and f 2M2.

(a) If ˛ D�a ba�1

�2 B.Q/, then a0.f j˛/ D a

2a0.f / and ef j˛ D Qf j˛.

26

(b) ef .˛ˇ/ D ef j˛.ˇ/.

(c) If � 2 T , the maximal torus of G.Q/, then ef .˛�/ D ef .˛/.

(d) If ! is the Weyl element as above, then ef .˛.1C !// D 0.

(e) If z0 2 H, then

ef .1/ D

Z i1

z0

Cf j.1 � !/.z/dz � z0 � a0�f j.1 � !/

��

Z !z0

z0

f .z/ dz:

Proof. (a) We compute

.f j˛/.z/ D j.˛; z/�2f .˛ � z/ D a2f�azCba�1

�D a2f .a2z C ab/I

thus a0.f j˛/ D a2a0.f /. Moreover,

.ef j˛/.z/ D .f j˛/.z/ � a0.f j˛/ D a2f .˛z/ � a2a0.f / D . Qf j˛/.z/:

(b) By definition,ef .˛ˇ/ D D

�f j.˛ˇ/; 1

�D D

�.f j˛/jˇ; 1

�D ef j˛.ˇ/:

(c) With � D�t1=2

t�1=2

�, we have by definition

D�f j.˛�/; s

�D

Z i1

0

. ef j˛�/.z/ys�1 dz D

Z i1

0

�B.f j˛/j��.z/ys�1 dz:

By part (a) we know B.f j˛/j� D .ef j˛/j� , so this equalsZ i1

0

.ef j˛/.� � z/j.�; z/�2ys�1 dz D

Z 10

.ef j˛/.� � z/y.z/s�1 d.� � z/:

As j.�; z/ D t; � � z D tz, this isZ i1

0

.ef j˛/.z/y.��1z/s�1 dz D t1�sZ i1

0

.ef j˛/.z/ys�1 dz D t1�sD.f j˛; s/I

both of these integrals converge by proposition 2.2.1(c), and setting s D 1 gives the result.

(d) This is immediate from the functional equation D.f j˛; s/CD�f j.˛!/; 2 � s

�D 0:

(e) By proposition 2.2.2 (b), we have equality for z0 D i with s D 1. The general result now holdsbecause the derivative of the right-hand side with respect to z0 is zero.

We observe that part (e) of proposition 2.2.4 solves our problem, and so we will restate it as atheorem.

Theorem 2.2.5. Let f and �f be as above; then the period of f is

�f .!/ D �D.f; 1/ D1

2�iL.f; 1/:

27

The problem of evaluating periods is thus reduced to finding special values of L-functions.

In closing, we record a few other properties of the cocycle �f .

Proposition 2.2.6 ([Ste12], proposition 2.3.3). Let f and �f be as above, ˛; ˇ 2G.Q/, and z0 2 H. Then:

(a) �f .˛ˇ/ D �f j˛.ˇ/C �f .˛/.

(b) Let ˛ D�a bc d

�and suppose c � 0; then

�f .˛/ D

(aca0.f /C

dca0.f j˛/ � ef

�c�1 a

c

�if c > 0;

bda0.f / if c D 0:

(c) �f�1 11

�D a0.f /.

Proof. (a) This is proven by the computation in section 2.1 (pp. 18–19).

(b) Let B D B.Q/ denote the standard Borel subgroup of upper-triangular matrices, and recall theBruhat decomposition

G.Q/ D Ba

B!BI

the case c D 0 now corresponds to the case ˛ 2 B, and c > 0 to the case ˛ 2 B!B. In the firstcase, we may write

�f .˛/ D

�Z ˛z0

z0

Qf .z/ dz C a0.f /.˛z0 � z0/

�� z0 � a0

�f j.˛ � 1/

�C

Z i1

z0

Cf j.˛ � 1/.z/dz:

As y.˛z/ D a2y.z/, we see that ˛z0 ! 1 as z0 ! 1, and so taking the limit both integrals onthe right hand side vanish and be obtain

�f .˛/ D limz0!i1

�a0.f /.˛z0 � z0/ � z0 � a0.f j˛/C z0 � a0.f /

�:

By proposition 2.2.4(a) we know that a0.f j˛/ D a2a0.f /, and so

�f .˛/ D limz0!i1

a0.f /

�a2z0 C ab � z0 � a

2z0 C z0

�D limz0!i1

aba0.f / D aba0.f /:

If c D 0 then a D d�1, and the claim is proven.

When c > 0, we use the decomposition

˛ D

�c�1 a

c

���1

1

��1 c�1d

1

�2 B!B

to write

�f .˛/ D �f .ˇ/C �f jˇ .!/C �f j.ˇ!/. /;

28

where ˇ D

�c�1 a

c

�and D

�1 c�1d

1

�. We know each of these values by parts (b) and (c):

�f .ˇ/ Da

ca0.f /I

�f jˇ .!/ D �D.f jˇ; 1/ D �ef .ˇ/I

�f j.ˇ!/. / Dd

ca0.f j.ˇ!// D

d

ca0�.f j˛/j �1

�Dd

ca0.f j˛/:

Note that in the last calculation, we have used the facts that ˇ! D ˛ �1 and that �1 is uppertriangular.

(c) This is an immediate corollary of part (b).

We remark that proposition 2.2.6(b) furnishes another proof of theorem 2.2.5.

2.3 Simplifications

We now have, in theorem 2.2.5, an effective way to calculate the periods of modular forms. Asjustified by our remarks in section 1.3, we may restrict our attention to the space E2 of Eisensteinseries of weight 2, of all levels, i.e., the complement of S2 in M2, as the periods of cusp forms maybe computed classically. Indeed, this is implied by the relation �fCg D �f C�g which furthermoreimplies that it is enough to compute the values of �f on ! 2 G.Q/ for all f in a generating setff g of E2.

We can reduce the problem still further by restricting our attention to those Eisenstein series whoseconstant term (in the Fourier expansion) is rational: for any subfield K of C, let E2.K/ be the setof E 2 E2 such the constant term of the q-expansion of E at every cusp lies in K; it may be shown([Ste12], p. 55) that E2.K/ is a K-vector space and that E2.K/ Š E2.Q/˝QK. Therefore it sufficesto construct a basis for E2.Q/.The connection between special values of L-functions and constant terms of Fourier series of Eisen-stein series has been demonstrated above (for instance, in propositions 2.2.3 and 2.2.6), and so weare motivated to find a basis for E2.Q/ whose Fourier coefficients may be computed easily. We willachieve this in two steps: first, by constructing a spanning set f}ag of Eisenstein series in a wayanalogous to the construction of E2k.zIN/ in section 1.2, then by obtaining a new basis f�xg whoseFourier series we then explicitly compute.

As we remarked in section 1.2, the exact constructions cannot be carried over to the weight 2case: this is because the series defined in equation (1.3) diverges if we put k D 1. However, byreplacing the summand j. ; z/�2k with j. ; z/�2jj. ; z/j�s for s 2 C, one obtains a function which(for sufficiently large R.s/) converges absolutely, and meromorphically extends to a function whichis analytic at s D 0. As such, we caution that while our investigation of the Eisenstein seriesof weight 2 bears many similarities to our calculations from section 1.2, it will also hold severalimportant distinctions.

29

2.4 An ancillary basis of Eisenstein series

In this section we construct a set fGag of functions in a way which mimics the classical constructionof the Eisenstein series we saw in section 1.2. There, we obtained a modular form by summingover negative exponents of lattice points of the lattice ƒ.z/ WD Z C zZ � C (except zero); here,we will perform roughly the same operation, summing this time over various elements of certainsublattices of ƒ.z/.

The functions Ga that we will construct are not quite Eisenstein series, but may be combined toyield a spanning set f}ag of Eisenstein series of weight 2 (and all levels). Because our previousconstruction is not valid when the weight is 2, we instead work with the analytic continuation of aclosely-related function.

Let a be the row vector .a1; a2/ 2�Q=Z

�2, and for z 2 H; s 2 C, put

Ga.z; s/ DX

.m;n/2Z2

0 1�.mC a1/z C .nC a2/

�2 � 1ˇ.mC a1/z C .nC a2/

ˇs ;the prime as usual indicating that the possibility .m; n/ D .a1; a2/ D 0 is excluded. This convergesabsolutely in s in some half-plane, and continues to a function which is holomorphic in s at s D 0([Hec27], pp. 206–7). For any a 2 .Q=Z/2, put Ga.z/ D Ga.z; 0/

Proposition 2.4.1 ([Ste12], proposition 2.4.1(c)). Let a 2 .Q=Z/2nf0g; then

Ga.z/ D }a.z/CG0.z/;

where }a.z/ D }.a1zCa2I z/ and }.zI �/ is the Weierstrass elliptic function associatedto the lattice ƒ.�/ D ZC �Z (see appendix A)

Proof. If a ¤ 0 we have

Ga.z; s/ DX

.m;n/2Z2

1�.mz C n/C .a1z C a2/

�2 � 1ˇ.mz C n/C .a1z C a2/

ˇs :The summand corresponding to .m; n/ D 0 is 1

.a1zCa2/2�

1ja1zCa2j

s , and by the identity

1�.mz C n/C .a1z C a2/

�2 � 1ˇ.mz C n/C .a1z C a2/

ˇsD

1

.mz C n/2�

1

jmz C njsC

1�.mz C n/C .a1z C a2/

�2 � 1ˇ.mz C n/C .a1z C a2/

ˇs�

1

.mz C n/2�

1

jmz C njs;

valid for .m; n/ ¤ 0, we obtain

Ga.z; s/ D

� X.m;n/2Z2

0 1

.mz C n/2�

1

jmz C njs

�C

1

.a1z C a2/2�

1

ja1z C a2j2

C

X.�m;�n/2Z2

0�

1�.a1z C a2 � .mz C n/

�2 � 1ˇ.a1z C a2 � .mz C n/

ˇ2 � 1

.mz C n/2�

1

jmz C njs

�:

30

The first summation is precisely G0.z; s/, and the remaining terms contribute precisely }a.z/, asdefined in appendix A. Taking the residue at s D 0 gives

Ga.z/ D }a.z/CG0.z/;

as claimed.

It may be shown ([Hec27], loc. cit.), for D�a bc d

�2 �.1/ and arbitrary a 2 .Q=Z/2, that

Ga. � z/ D j. ; z/2Ga .z/;

where a D

�aa1 C ca2ba1 C da2

�is ordinary matrix multiplication. For 2 �.N/ and a 2

�1NZ=Z

�2, then

we may compute

a D�a1

N; a2

N

� �1C a0N b0N

c0N 1C d 0N

�D�a1=N C .a

0a1 C c0a2/; a2=N C .b

0a1 C d0a2/

�D a;

and thus

Ga. � z/ D j. ; z/2Ga.z/:

The functions Ga.z/ are not quite holomorphic in z; however, we have

Proposition 2.4.2 ([Ste12], proposition 2.4.1 (b),(d)). Retaining notation from above,one has

(a) For a 2 .Q=Z/2 n f0g, the function

z 7! Ga.z/C2�i

z � Nz

is holomorphic in H.

(b) The set

f}a.z/ W a 2�1NZ=Z

�2n f0gg

generates E2.N /.

Thus we may use the functions }a (and, consequently, Ga) to build a basis of the space E2.N /.However, while more intuitive to construct, these functions have the drawback of being less suitedto our analytic methods. We will therefore, in the next section, use the functions fGag to buildanother basis of Eisenstein series, whose periods (as defined in section 2.1) we will be able tocompute explicitly.

2.5 Fourier coefficients of Eisenstein series

In this section, we will use the series Ga from the previous section to obtain a new basis of Eisensteinseries, whose Fourier coefficients are computed in proposition 2.5.1.

31

Every element a of�Q=Z

�2may be canonically written

�a1

N; a2

N

�for some minimal positive integer

N and ai 2 Z. Thus we are led to consider the groups CN WD1NZ=Z.

Let x D .x1; x2/ 2 C2N �

�Q=Z

�2, and define the character x W C

2N ! C� by��

a1

N; a2

N

�; x

�7�! exp

�2�i.a2x1 � a1x2/

�:

Such characters are obviously distinct (i.e., x ¤ y for x ¤ y), and therefore by cardinality

considerations exhaust the entire dual group of C 2N , which is isomorphic to C 2N itself as CN isfinite. Now, define

�x.z/ D1

.2�N/2

Xa2C2

N

x.a/Ga.z/:

We elucidate this definition: fix z 2 H, and put

ˆz.x/ WD �x.z/; gz.x/ WD Gx.z/;

to obtain functions C 2N ! C�, where CN D1NZ=Z as above. Then by the identity .�x; a/ D

.x; �a/ we may more compactly write

ˆz.x/ D1

.2�/2N 2

Xa2C2

N

gz.a/�a; x

�D

1

4�2

�1

N 2

Xa2C2

N

gz.a/�� a; �x

��D

1

4�2bgz.x/;

where Of is the ordinary Fourier transform on C 2N . By the Fourier inversion formula, therefore, wehave

gz.x/ DXa2C2

N

bgz.a/�a; x� D 4�2 Xa2C2

N

ˆz.a/�a; x

�;

and thus

Gx.z/ D 4�2Xa2C2

N

x.a/�a.z/:

If x ¤ 0, then �x is a weight 2 modular form of level N : indeed, the modularity condition is satisfiedby each summand (as shown above), and holomorphy is a consequence of its Fourier expansion(proposition 2.5.1, below), as is nonvanishing at the cusp. The function �0 is not holomorphic,though it still satisfies the modularity condition.

A basis for E2.N /, N > 1, may therefore a priori be obtained from the set f�x W x 2 .1NZ=Z/2g; in

fact, one may do this explicitly ([Ste12], pp. 60–61). Roughly speaking, a spanning set for E2��.N/

�is obtained by taking those �x for which the least common multiple of the denominators of theentries x is exactly N .

The functions �x have certain advantages over the functions introduced in the last chapter, someof which are established in [Ste12], and which we will state here.

Proposition 2.5.1 ([Ste12], proposition 2.4.2). With notation as above, and B2.x/ Dx2 � x C 1

6the second Bernoulli polynomial, one has:

32

(a) For x D .x1; x2/ 2 .Q=Z/2nf0g, we have the Fourier expansion (with q D q.z/ Dexp.2�iz/ as before)

�x.z/ D1

2B2.x1/�

Xk�x1 mod 1k2Q>0

k

1XmD1

q�m.kzCx2/

��

Xk��x1 mod 1

k2Q>0

k

1XmD1

q�m.kz�x2/

�:

(b) The function z 7! �0.z/ Ci

2�.z�Nz/is holomorphic in z 2 H and its Fourier

expansion coincides with the formula in part (a):

�0.z/Ci

2�.z � Nz/D

1

12� 2

1XmD1

1XkD1

kq.mkz/ D1

12� 2

1XnD1

�Xd jn

d

�q.nz/

D1

12� 2

1XnD1

�.n/q.nz/;

where �.n/ is the sum-of-divisors function.

The second part of this proposition immediately yields

Corollary 1. If G2.z/ is the quasi-holomorphic Eisenstein series from section 1.2,then

G2.z/ D 12�0.z/:

With our Fourier coefficients in hand, we are now ready to compute our periods: for x 2 Q=Z,define two functions

Z.s; x/ D

1XnD1

q.nx/

ns; and �.s; x/ D

Xk�x mod 1k2Q>0

1

ks:

Proposition 2.5.2 ([Ste12], proposition 2.5.1). Let x 2 .Q=Z/2 n f0g. One has:

(a) D.�x; s/ D �i�.s/.2�/�s�Z.s; x2/�.s � 1; x1/CZ.s;�x2/�.s � 1;�x1/

�:

(b) D.�x; 1/ D B1.x1/B2.x2/C1

2�i

�ıx10 log j1� q.x2/j � ıx20 log j1� q.x1/j

�, where

ıij is the Kronecker delta function and B1.x/ D x �12

the first Bernoulli polyno-mial.

Proof. [Ste12], pp. 67–69.

2.6 Periods of basis elements

Proposition 2.5.2(b) already allows us (by theorem 2.2.5) to compute our periods exactly, andproposition 2.2.3(b) relates the values of these periods to the special values of the associated L-functions. Summarizing, we obtain our main result:

33

Theorem 2.6.1. Let �x; x ¤ 0 be the Eisenstein series from section 2:5 and letB1.x/ D x � 1

2; B2.x/ D x2 � x C 1

6be the first and second Bernoulli polynomials,

respectively. Then if L.�x; s/ is the associated L-function, one has

L.�x; 1/ D ıx20 log j1 � q.x1/j � ıx10 log j1 � q.x2/j � 2�iB1.x1/B2.x2/;

where ıij is the Kronecker delta function, and so the period of �x is

��x.!/ D

1

2�i

�ıx20 log j1 � q.x1/j � ıx10 log j1 � q.x2/j

�� B1.x1/B2.x2/:

Combining theorem 2.6.1, the simplifications of section 2.3, and the linearity relation �fCg D

�f C �g , we obtain

Corollary 1. Let E 2 E2.N /, and write

E DXx2C2

N

cx�x;

where cx 2 C and CN D1NZ=Z as before. Then the period of E is

�E .!/ DXx2C2

N

cx��x.!/:

With this corollary, we finally achieve the goal we outlined in section 1.3: the pairing . ; / on Y.�/which we desire is precisely that which is defined on the contour f0; i1g 2 H1

�Y.�/IC

�by�

f0; i1g; f�7�! �f .!/;

and whose values on other homology classes are obtained by the discussion of appendix C.

This having been achieved, we will in the next chapter transfer this result to the adelic world;after reviewing the necessary background, we will see that theorem 2.6.1 (and its corollary) can berestated in terms of the modern theory of automorphic forms.

34

Chapter 3

The adelic picture

To understand periods of modular forms from a more modern perspective, we begin first by explor-ing how the upper half-plane and various functions on it translate to the adelic picture.

In section 3.1, we see how the classical setting for our investigation is canonically identified with aquotient space of adelic algebraic groups. In section 3.2 we review the measure theory of SL2.R/,which we then use in section 3.3 to transfer integrals on contours in H to integrals over double-cosetspaces in G.Q/nG.A/=KS . This allows us to define the period of a modular form purely adelically.

In section 3.4 we see how classical Eisenstein series are realized as elements of particular inducedrepresentations, the automorphic Eisenstein series, which finally allows us to define the period ofan automorphic Eisenstein series independently of the classical setting.

3.1 The classical-adelic correspondence

The basic reference is [Cas12].

We recall first the definition of the adele ring A of Q. This is the restricted product of all valuationsv of Q with respect to the open compact subgroups Zp:

A D R �Yp

0

QpI

by convention we put Q1 D R. That is, an element of A is a sequence of the form a D

.a1; a2; a3; a5; : : :/, with all av 2 Qv and all but finitely many ap 2 Zp. This is a topologicalgroup under the restricted product topology: a basis for the topology consists of open sets whichare sequences .U1; U2; U3; U5; : : :/ of open sets U1 � R and compact open Up � Qp, with all butfinitely many Up equal to Zp.

The fundamental theorem of arithmetic implies that the diagonal embedding a 7! .a; a; a; : : :/ ofQ into A is well-defined; where there will be no confusion, we will identify Q with its image in Aunder this inclusion. We will also sometimes write Afin for the finite adeles, that is, the restrictedproduct over the finite places:

Afin D

Yp

0

Qp:

35

The group of units A� of the adeles (known as the idele group) is similarly defined as the restrictedproduct of all completions of Q� with respect to the subgroups Z�p; thus an idele may be writtena D .a1; a2; a3; a5; : : :/, with all av 2 Q�v and all but finitely many ap 2 Z�p. We note that thetopology on the ideles is not the subspace topology of A� in A.

The diagonal inclusion Q ,! A exhibits Q as a discrete additive subgroup of A: to see this, wesimply note that the open set

.�1; 1/ �Yp

Zp

contains only one rational number, i.e., zero. In fact, it may be shown that Q� is dense in A�: thisis one of the most fundamental results about the ideles, and we will prove it here.

Proposition 3.1.1 ([Cas12], proposition 2.2). The image of Q� in A� under thediagonal embedding is dense. In particular,

A� D Q� ��R>0 �

Yp

Z�p�:

Proof. Let a D .a1; a2; a3; a5; : : :/ 2 A�, and suppose S is the (finite) set of primes for whichap … Z�p. If p 2 S , then because ap 2 Q�p by assumption we know ap 2 p

npZ�p for some integernp, and of course pnp 2 Z�q for all q ¤ p. Thus� Y

p2S

p�np

�a 2 R� �

Yp

Z�p;

and the assertion is proven.

Let G D SL2 as above; the group G.A/ coincides with the restricted product of the groups G.Qv/with respect to the groups G.Zp/, and we similarly have the diagonal inclusion G.Q/ ,! G.A/which will lead us to identify G.Q/ with its image in G.A/. There is also a natural inclusion

G.R/ G.A/g .g; 1; 1; : : :/

where 1 denotes the identity matrix.

As G is simply connected, proposition 3.1.1 easily generalizes to the theorem of strong approxima-tion, which we will state here.

Theorem 3.1.2 ([Cas12], theorem 2.3). Let Kfin � G.Afin/ be any compact opensubgroup; then

G.A/ D G.Q/ ��G.R/ �Kfin

�:

Equivalently, G.Q/G.R/ is dense in G.A/.

Proof. See [Cas12], p. 3.

36

For every finite set of (finite) primes S , there is the natural surjection G.A/! G.A/=KS;fin, where

KS;fin D 1 �Yp2S

Kp �Yp…S

Kp;

with Kp � G.Zp/ a compact subgroup for all p and Kp D G.Zp/ for all p … S . Taking the quotienton the left-hand side by G.Q/ (more precisely, by its image in G.A/ via the diagonal inclusion), wecompose this map with the inclusion G.R/ ,! G.A/ to obtain a function

G.R/! G.Q/nG.A/=KS;fin:

Theorem 3.1.2 implies that this descends to a one-to-one correspondence

Q�S W �SnG.R/=K1��! G.Q/nG.A/=KS

where KS D K1 �KS;fin and

�S D fg 2 G.Q/ W .g; g; g; : : :/ 2 G.R/ �KS;fing:

By the identification of G.R/=K1 with H from section 1.1, this implies that the modular surface�SnH may be identified with the adelic double quotient G.Q/nG.A/=KS .

We will write �S for the inverse of the composition of this identification with Q�S ; that is, withz D x C iy 2 H, we have a one-to-one correspondence

G.Q/nG.A/=KS �SnH

G.Q/� .y1=2;1;1;:::/ .xy�1=2;1;1;:::/

.y�1=2;1;1;:::/

�KS �Sz

�S

In particular, if N � 1 is an integer, then write

N DYp

p�.p/; �.p/ D 0 for all p … S;

and put

Kp D

(1C p�.p/G.Zp/ if p 2 S;

G.Zp/ if p … S:

Then�S D fg 2 G.Q/ W g 2 Kp 8p and g � 1 mod p�.p/G.Zp/ for p 2 Sg D �.N/:

We may now translate with ease between the classical and the adelic setting.

3.2 Integration on SL2.R/Now that we have the correspondence �SnH Š G.Q/nG.A/=KS , we will investigate how the mea-sures on these spaces are related. We begin by reviewing the measure theory of G.R/; the primaryreference is [Lan12], p.37–45.

37

Let G be a locally compact topological group, and let Cc.G/ be the space of continuous, compactly-supported, complex-valued functions on G. We recall the notion of Haar measure: a left Haarmeasure on G is a positive measure dLg such that, for all f 2 Cc.G/, one hasZ

G

f .xg/dLg D

ZG

f .g/dLg

for every x 2 G; right Haar measure is defined analogously and will be denoted dRg. It may beshown that Haar measure is unique up to multiplication by a constant, and furthermore that thereexists a homomorphism ıG W G ! R such that

dRg D ıG.g/dLg:

This homomorphism is called the modulus character of G, and the group G is called unimodularif ıG � 1; that is, if left and right Haar measure coincide, in which case we will write dLg D dRg Ddg.

We will be interested in the case whenG D SL2.R/ andK D SO2.R/, both of which are unimodular.If f 2 Cc.G/, define for gK 2 G=K the function

f K.gK/ D

ZK

f .gk/ dk:

It may be shown not only that f K 2 Cc.G=K/, but also that f 7! f K yields a surjection Cc.G/�Cc.G=K/. We have (loc. cit., Theorem 1)

Lemma 3.1. There is a unique, left G-invariant measure d Pg on G=K such that, forall f 2 Cc.G/, one has Z

G

f .g/dg D

ZG=K

f K.gK/d Pg:

We remark that, if f 2 Cc.G/ is right K-invariant, then we may normalize Haar measure on K sothat

vol.K/ D

ZK

dk D 1;

hence f K.gK/ D f .g/, and thus f ! f K establishes a natural one-to-one correspondencebetween right K-invariant functions f 2 Cc.G/ and functions in Cc.G=K/.

Now: let B be the subgroup of upper-triangular matrices in G with positive diagonal entries (incontrast to its usage above). The map .b; k/ 7! bk gives a homeomorphism of B �K onto G whichis also an isomorphism of groups. Moreover, if db; dk are respective left Haar measures on B;K,then db and dk may be scaled so that vol.K/ D 1 and, for all f 2 Cc.G/, one hasZ

G

f .g/dg D

ZB

ZK

f .bk/ dk db D

ZK

ZB

f .bk/db dk;

the second equality holding by Fubini’s theorem (note that we have not commuted the argumentbk of f ).

38

We may further decompose B D AN , where A is the subgroup of diagonal matrices and N thesubgroup of upper-triangular unipotent matrices, both of which are unimodular:

A D f

�a

a�1

�W a 2 R>0g; N D f

�1 n

1

�W n 2 Rg:

A quick calculation shows thatANA�1 D N;

from which we conclude two things: first, that there is a homeomorphism A � N ! B given by.a; n/ 7! an, and therefore that the Haar measures da;dn on A and N , respectively, may be scaledso that

db D da dn

and therefore thatdg D da dndk:

Second, there exists a homomorphism from A to the group AutC .N / of continuous automorphismsof N (namely, the conjugation-by-a map), and because

AutC .N / Š AutC .R;C/ Š R>0;

we see that there is a continuous group homomorphism ˛ W A! R>0 such thatZN

f .ana�1/dn D ˛.a/�1ZN

f .n/dn;

and thus ZG

f .g/dg D

ZK

ZA

ZN

˛.a/�1f .nak/dn da dk: (3.1)

The choice of notation is deliberate: the only positive root of the algebraic group SL2.R/ is

˛

�a

�a

�!D a2;

which (as we will see below) is precisely our function ˛.

Finally, if ıB is the modulus character from before, then one also has ı.b/ D ı.an/ D ˛.a/; thiswill allow us to compute these measures in co-ordinates.

First, write B D NA; these co-ordinates (rather than B D AN ) will be more useful for our purposes.Write

A D f

�u

u�1

�W u 2 R>0g; N D f

�1 x

1

�W x 2 Rg;

and y D u2, so that �1 x

1

��u

u�1

�� i D x C iy:

Our work above implies thatZA

ZN

f .an/ dnda D

ZN

ZA

˛.a/�1f .na/da dn;

39

and therefore the calculation�u

u�1

��1 x

1

�D

�1 u2x

1

��u

u�1

�implies

˛.a/ D ˛

�u

u�1

�!D u2:

If dCu is the ordinary Lebesgue measure on R (i.e., Haar measure on the additive group R), thendy D 2u dCu, and so the identification

x C iy 2 H !�1 x

1

��u

u�1

�D na 2 NA D B

yields an equality of measures

dx dy

y2D

dx.2udCu/

.u2/2D2 dx d�u

˛.a/;

where d�u D dCuu

is Haar measure on the (multiplicative) group R>0. Finally, identifying dx !dn;da ! d�u, we have in these co-ordinates that

dx dy

y2on H !

2dnda

˛.a/on G=K D NA;

and thus the measure d Pg from lemma 3.1 satisfies for every f 2 Cc.G=K/ZG=K

f .gK/d Pg D

ZA

ZN

2˛�1.a/f .an/dn da:

Putting this all together, we have

Theorem 3.2.1. Let F W H ! C be continuous with compact support, and definef W G=K ! C via f .gK/ WD F.g � i/. ThenZ

HF.z/

dx dy

y2D

ZG=K

f .gK/d Pg D

ZA

ZN

2˛�1.a/f .an/dn da:

As we have seen (section 1.1), for appropriate arithmetic subgroup � the quotient space �nH isidentified with a subset of H (namely, some fundamental domain D). As such, the measure on �nHis naturally identified with the measure on H: for any F 2 Cc.�nH/, one hasZ

�nHF.�z/d��nH.z/ D

ZHf .z/1D.z/

dx dy

y2D

ZD

f .z/dx dy

y2;

where f . � z/ WD F.�z/ and 1X denotes the characteristic function of X .

More generally, for any unimodular G and discrete closed H � G, Haar measures dg;dh;d Pg maybe normalized so that for any f 2 Cc.G/ one hasZ

HnG

�ZH

f .hg/dh

�d Pg D

ZG

f .g/dg:

40

In particular, letting G D B D AN and H D �0 D � \ B, we see A \ � D f1g and thus �0 � N .Consequently, �0nB Š A � �0nN , and we haveZ

B

f .b/db D

Z�0nB

� X 2�0

f . b/

�d Pb

D

ZA

Z�0nN

� X 2�0

f . an/

�d Pnda

D

ZA

ZN

f .an/da dn

D

ZA

ZN

˛�1.a/f .na/da dn:

Combining this observation with theorem 3.2.1 yields

Corollary 1. Let F 2 Cc.�nH/ and let f W �nG=K ! C be defined f .�gK/ D

F��.g � i/

�. Abusing notation by writing F.z/ D F.�z/ and n D �0n, we haveZ

�nHF.z/

dx dy

y2D

ZA

Z�0nN

2˛�1.a/f .na/dnda:

In particular, Haar measure d Pn on �0nN is identified with Haar measure dn on N

in our co-ordinates.

From this, we deduce

Corollary 2. Let D be a fundamental domain for the action of � � G.Z/ on H, letF 2 Cc.D/, extend F to H via F. � z/ D F.z/, and put f .�gK/ WD F.g � i/; thenZ

D

F.z/dx dy

y2D

Z�0nB

f .�0bK/ d Pb

D

ZA

Z�0nN

2˛�1.a/f .na/dnda:

In the next section, we will restrict our attention to integration over particular contours in H.

3.3 Integration on adelic groups

Let us consider now what occurs in the adelic world when we follow the approach of Stevens.Retaining notation from section 3.1, we begin with the simplest case when S D ; and so �S D �.1/;write K for KS D K; D K1 �

Qp G.Zp/.

Recalling from section 1.1 the fundamental domain D for the action of PGL2.Z/ Š �.1/=f˙1g onH, whose interior may be identified with the affine curve Y.�/, we observe that the geodesic f0; i1gconsists of the union of two contours, namely

fi; i1g � D and f0; ig � ! �D;

41

where ! D��1

1

�as before. Under the correspondence �;, we have

fi; i1g D fiy W y � 1g ! fG.Q/� .y1=2;1;1;:::/

.y�1=2;1;1;:::/

�K; 2 G.Q/nG.A/=K; W y > 1g;

and similarly

f0; ig D fiy W 0 � y � 1g ! fG.Q/� .y1=2;1;1;:::/

.y�1=2;1;1;:::/

�K; 2 G.Q/nG.A/=K; W 0 < y < 1g:

To simplify these expressions, we introduce the Harish-Chandra homomorphism ([Art05],p. 22) associated to the subgroup B of G, and which in our case is a surjective homomorphismHB W T .A/! R defined

HB

�a

a�1

�!D log jaj;

where T .A/ � G.A/ is the maximal torus and j � j DQv j � jv is the adelic absolute value.

It is apparent from our work above that the only cosets corresponding to the contour f0; i1g under�; have diagonal representatives, and therefore that the restriction of �; to T .A/ surjects ontof0; i1g. We will abuse notation by continuing to write �; for the map

T .Q/nT .A/=KT�!f0; i1g;

where KT D K; \ T .A/. We may now write, for instance, that

fi; i1g D f�;�T .Q/tKT

�W HB.t/ > 0g:

We observe that HB is well defined on the coset space, as HB�T .Q/

�D HB.KT / D 1.

We also note that, if f .z/ is a cusp form, we have by equation (2.2) thatZ i1

0

f .z/d z D

Z i

0

f .z/dz C

Z i1

i

f .z/dz D

Z i

i1

.f j!/.z/dz C

Z i1

i

f .z/ dz

D

Z i1

i

f j.1 � !/.z/dz; (3.2)

(coinciding with �f .!/), and so we can restrict our attention to the contour fi; i1g. Of course, forany modular form for �.1/ we have f j! D f and so (3.2) tells us only that the periods of all cuspforms of weight 2 for �.1/ is 0, which is to be expected as M2

��.1/

�D 0.

For general S and KS DQp Kp, we have

Lemma 3.2. Let �.N/ � G.Z/; N � 2 be a congruence subgroup and let KS;fin DQp Kp be its image in the finite adeles; put KS D K1 �KS;fin as before. Then the

image of f�T .Q/tKT

�W HB.t/ > 0g under �S is the contour fi; i1g in �SnH, and the

mapping is one-to-one.

Proof. Let

�a

a�1

�2 T .A/, and write a D .a1; a2; a3; a5; : : :/. By theorem 3.1.2, there exists a

matrix g 2 T .Q/ (identified, as usual, with its image in T .A/) such that

g

�a

a�1

�2 T .R/ �

�T .Afin/ \

Yp

Kp

�;

42

and so we may assume without loss of generality that ap 2 Kp for all p. Thus the image of�a

a�1

�in T .Q/nT .A/=KT is the same as the image of

�.a1; 1; 1; : : :/

.a�11 ; 1; 1; : : :/

�. Finally,

since �I 2 T .Q/, it suffices to assume a1 > 0, and we have that

f0; i1g D �S�fG.Q/

�.y1=2; 1; 1; : : :/

.y�1=2; 1; 1; : : :/

�KS W y > 0g

�:

The remaining assertion is a consequence of the fact that two double cosets are the same, i.e.,

T .Q/

.y1=21 ; 1; 1; : : :/

.y�1=21 ; 1; 1; : : :/

!KT D T .Q/

.y1=22 ; 1; 1; : : :/

.y�1=22 ; 1; 1; : : :/

!KT ;

if and only if there exist t 2 T .Q/; k 2 KT such that .y1=21 ; 1; 1; : : :/

.y�1=21 ; 1; 1; : : :/

!D t

.y1=22 ; 1; 1; : : :/

.y�1=22 ; 1; 1; : : :/

!k:

By considering the finite places, it is obvious that for this equality to hold we must have

k D

�.k; 1; 1; : : :/

.k�1; 1; 1; : : :/

�;

which lies in KS if and only if k D ˙1. If k D �1, then we must have �I 2 KT � KS and so�I 2 �.N/, which is absurd when N > 1. It follows that k D 1, and the claim is proven.

Recall the slash operator from chapter 1; using the same notation and terminology, we will definethe slash operator on the space of functions on G.A/ (and its quotients) via

.F j /�g/ WD j. 1; z/

�2F. g/

for 2 G.A/, where z D x C iy and x; y are obtained by the decomposition

g1 D

�y1=2 xy�1=2

y�1=2

�k1

and k1 2 K1. The decomposition of g1 is obtained in the proof of lemma D.2.

Lemma 3.3. Let f 2 Cc.�nH/. ThenZ i1

0

f .z/dz D

Z i1

i

f .z/dz �

Z i1

i

.f j!/.z/dz:

Furthermore, if F W G.Q/nG.A/=KS ! C satisfies F.g/ D f .z/ (with the associationz $ g1 from above), then for the Weyl element ! one has

.F j!/.g/ D .f j!/.z/:

Proof. The first claim is an immediate corollary of equation (2.2), and the second follows fromlemma D.2.

43

Consequently, we can always restrict our attention to integration on the contour fi; i1g, and mayfreely use the slash operator in the adelic world. We will use the notation f! D f j.! � 1/ in theadelic context.

The same calculations from section 3.2 carry over mutatis mutandis to the problem of finding Haarmeasure in co-ordinates on T .A/. The real points T .R/ of the torus are identified with the groupA from that section; consequently, in the same co-ordinates, we have

dy on H ! 2˛.a/da on A:

Thus we obtain

Theorem 3.3.1. Retain notation from lemma 3.2, and let f 2 S2.�/ be a cusp formof weight 2. Then the period of f is

�f .!/ D

Z i1

i

f j.1 � !/.z/dz D

Z 11

f j.1 � !/.iy/ i dy

D

ZT.Q/nT.A/=KT

.f! ı �/.t/ 2i˛.t/dt;

where dt is Haar measure on the torus.

Generally, we may now compute periods of modular forms f 2M2 by the formula

�f .!/ D

Z !i

i

f .z/ dz � ia0�f j.! � 1/

�C

Z i1

i

Cf j.! � 1/.z/ dz

D

Z i1

i

Cf j.! � 1/.z/ dz � ia0�f j.! � 1/

�It remains only to express a0.f / in terms of adelic integrals; this is precisely

a0.f / D

ZN.A/

.f ı �/.an/dn;

and we have

Theorem 3.3.2. Let f 2M2 be a modular form of level N . Let KS be the imageof �.N/ in G.A/, let KT D T .A/ \ KS and let � D �S . Abuse notation by writingf! D f! ı � ; then the period of f is

�f .!/ D 2

ZT.Q/nT.A/=KT

�f!.t/ �

ZN.A/

f!.n/ dn

�˛.t/dt � i

ZN.A/

f!.n/dn;

where dt is Haar measure on T and dn is Haar measure on N .

3.4 Representations

Eisenstein series may be realized as elements of certain representations of particular matrix groups,and this perspective is, in many ways, the more modern way to consider them. We begin by

44

reviewing certain necessary terminology, taking as references [VJ08], pp. 260–262 and [Tro10],pp. 5–13.

Let G be a topological group. A representation of G is a pair .�; V / consisting of a vector spaceV and a group homomorphism � W G ! GL.V /; we often abuse notation by writing simply � or Vfor the whole representation. The representation is called continuous if the map

G � V V

.g; v/ �.g/v

is continuous, and is called smooth if, for every v 2 V , the stabilizer subgroup

StabG.v/ D fg 2 G W �.g/v D vg

is open in G. We also have the notion of an irreducible representation, which is a representation.�; V / of G such that V has no proper, nonzero subspace U satisfying �.G/U � U (a G-invariantsubspace).

A unitary representation is one for which V is a Hilbert space and �.g/ is a unitary operator forevery g 2 G; that is, �.g/ preserves the inner product on V and �.g/V is dense in V . If G isa locally profinite group with smooth representation � , then � is called admissible if, for everyopen subgroup K of G, the subspace

V K D fv 2 V W �.k/v D v for all k 2 Kg

is finite-dimensional.

Finally, we have the notion of the induced representation: if H is a closed subgroup of G and.�; V / is a smooth representation of H , then let VG be the space of functions f W G ! V such that

1. f .hg/ D �.h/f .g/ for all h 2 H , and

2. there exists an open subgroup K of G such that f .gk/ D f .g/ for all k 2 K.

Let �G W G ! VG be the right-translation map on VG :

�G.g/f .x/ D f .xg/:

Then we write I.�/ D IndGH��; V

�D .�G ; VG/, and call I.�/ the induced representation of G.

We will realize Eisenstein series as vectors in a particular induced representation of G.A/: for thesubsequent exposition, we refer to [Bro11].

Suppose 1 W B.A/ ! C is the trivial representation, and let I.1/ D IndG.A/B.A/.1/. Elements of VG.A/

are therefore functions f W G.A/! C such that

1. f .bg/ D f .g/ for all b 2 B.A/, and

2. there exists an open subgroup K of G.A/ such that f .gk/ D f .g/ for all k 2 K.

45

By Flath’s theorem ([Bum98], theorem 3.3.3 and �3.4), all representations we consider decomposeas restricted tensor products over all places of Q. In particular, we may write

1 DOv

1v;

the product taken over all places v, where .1v; Vv/ is a representation of G.Qv/ (Technically,.11; V1/ is an .sl2.R/;K1/-module, but the distinction will be unimportant for our purposes).We caution that, as with our previous restricted product constructions, additional restrictions areplaced on elements of the adelic representation (for instance, that every element lie in a Kv-fixedsubspace for all but finitely many places v); however, we will not construct elements of the adelicrepresentation from local elements, and so we ignore these issues here.

As such, every f 2 I.1/ satisfies

f 2

�Ov2S

I.1v/

��Ov…S

f 0v

�;

where f 0v .kv/ D 1 for all kv in the maximal compact subgroup Kv of G.Qv/, and S is a finiteset of places (dependent on f ); we will retain our notation K1 D SO2.R/;Kp D G.Zp/ fromearlier. Extend the domain of the modulus character ıB.b/ from earlier to all of G by the Iwasawadecomposition: ıB.bk/ D ıB.b/ for all k 2 K. Then we may define, for any s 2 C, the function onG.A/

fs.g/ D fs.bk/ D ıB.b/s=2f .bk/

and its corresponding Eisenstein series

E.gI s; f / DX

2B.Q/nG.Q/

fs. g/: (3.3)

Here, as before, we understand elements of G.Q/ as elements of G.A/ under the diagonal inclusion.

The choice of terminology is apt, as this new definition indeed supersedes the old one. One particularelement of our representation space is the element f D

Nv fv given by

fv.gv/ D

(j.k1; i/

�2k if v D1;

1 if v <1;

where g D .g1; g2; g3; : : : ; / D .b1k1; b2k2; b3k3; : : :/ as before. This function is obviously leftB.A/-invariant, and is right-invariant under translation by elements of 1 �

Qp G.Zp/. Writing

b1 D

�y1=2 xy�1=2

y�1=2

�; so that ıB.b1/ D y;

and b1 D b0k0, one has

fs. g1/ D ıB. g1/s=2f . g1/ D ıB.b

0k0k1/s=2f .b0k0k1/

D ıB.b0/s=2j.k0k1; i/

�2kD ıB.b

0/s=2j.k0; k1 � i/�2kj.k1; i/

�2

D ıB.b0/s=2j.k0; i/�2kj.k1; i/

�2k;

46

where we have used both the cocycle condition and the fact that K1 is the stabilizer of i underthe action of G.R/. With z D x C iy and D

�a bc d

�, one computes further

g1 D

�a b

c d

��y1=2 xy�1=2

y�1=2

�k1

D

y1=2

jj. ;z/j

�acy2 C .ax C b/.cx C d/

�jj. ;z/j

y1=2

jj. ;z/j

y1=2

! cxCdjj. ;z/j

�cyjj. ;z/j

cyjj. ;z/j

cxCdjj. ;z/j

!k1 D b

0k0k1;

hence

ıB.b0k0k1/ D ıB.b

0/ Dy

jj. ; z/j2; j.k0; i/ D i

cy

jj. ; z/jCcx C d

jj. ; z/jD

j. ; z/

jj. ; z/j

and thus

fs. g1/ Dys=2

j.k0; i/2kj.k1; i/2kjj. ; z/jsD

ys=2

j. ; z/2kj.k1; i/2kjj. ; z/js�2k:

Setting the parameter s to 2k, we obtain

E.gI 2k; f / DX

2B.Q/nG.Q/

yk

j. ; z/2kj.k1; i/2kD

yk

j.k1; i/2k

X 2B.Q/nG.Q/

j. ; z/�2k;

and finally asj.k1; i/ D j.g1; i/j.b1; i/

�1D y�1=2j.g1; i/;

we seeE.gI 2k; f / D j.g1; i/

�2kX

2B.Q/nG.Q/

j. ; z/�2k :

The comparison with the classical Eisenstein series from section 1.2 is now clear, with the variableg taking the place of the complex variable z 2 H; all classical Eisenstein series similarly arise fromelements f of the representation space I.1/ by summing over translates of the associated functionsfs.

By an automorphic Eisenstein series we mean a function E.gI s; f / of the form (3.3) whichconverges absolutely for sufficiently large R.s/, and which satisfies the relation

E. gI s; f / D j. ; z/�sE.gI s; f /

when 2 G.Q/ (where z is associated to g1 as above).

We will close by constructing the Eisenstein series of level N (corresponding to the cusp at infinity)in this fashion. Recall the Eisenstein series of weight 2k and level N from section 1.2, defined

E2k.zIN/ DX

2�0.N/n�.N/

j. ; z/�2k :

WriteN D

Yp

p�.p/ DYp2S

p�.p/;

47

and for every prime p define

Kp.N / D f 2 G.Zp/ W ��� �

�

�mod NZpg:

Then Kp.N / D G.Zp/ if p … S , and Kp.N / D Kp.p�.p// for p 2 S . Define

f1.g1/ D j.k1; i/�2k

as before, andfp.gp/ D 1Kp.S/;

with 1X the characteristic function as before; clearly fp � 1 for p - N . For 2 G.Q/; g1 2 G.R/,we have

f . g/ D f�. g1; ; ; : : :/

�D f1. g1/

Yp

fp. /;

which is zero unless . g1; ; ; : : :/ 2 G.R/�Qp Kp.N /, and in this case equals f1. g1/. Because

�.N/ D G.Q/ \�G.R/ �

Yp

Kp.N /

�;

the same calculations as before now give

E2k.zIN/ D E.gI 2k; f /:

Finally, we observe that (with this same choice of f ) the function E.gI s; f / continues analyticallyto s D 0 ([Lan76], appendix IV), giving us another way to construct Eisenstein series of weight 2.

3.5 Conclusions

We have now demonstrated that the naıve definition of the period of a cusp form f .z/ as theintegral Z i1

0

f .z/dz;

while not well-defined for an arbitrary modular form, coincides with the definition of the cocycle�f .!/. Furthermore, we have shown that �f .!/ is not only well-defined for any f 2M2, but maybe computed explicitly on basis elements, and therefore supersedes the previous definition.

The contour integrals yielding the values of �f .!/ may then be transferred to the adelic setting,allowing us to define �f .!/ solely in terms of adelic integration. This has the added advantageof allowing us to define the period of automorphic Eisenstein series independently of the upperhalf-plane and classical modular forms, to wit:

Definition. Let E.gI s; f / be an automorphic Eisenstein series. Then the period ofE is Z

T.Q/nT.A/=KT

�E!.t/ �

ZN.A/

E!.n/dn

�dt � i

ZN.A/

E!.n/ dn;

where the notation is retained from section 3.3.

48

Not only have we answered (affirmatively) the question we posed in section 1.3, but we have alsogained the freedom and the power to investigate periods from the modern, adelic perspective ofautomorphic forms. As modern mathematics increasingly recognizes this to be the more naturalsetting for the investigation of (classical) modular forms, we hope that these results will sparefuture researchers the necessity of translating between the two settings.

49

50

Appendix A

Lattices and elliptic curves

The primary reference is [Mil90], pp. 41–47.

Recall the notion of a lattice in C, that is, an additive subgroup of the form ƒ D Z!1 C Z!2,where !1; !2 2 C (called the periods of the lattice) are R-linear independent. The linear spaceC=ƒ naturally inherits the topology of a torus, and admits a unique complex structure, and so weconsider it as a Riemann surface; indeed, the desire to understand all meromorphic functions onall Riemann surfaces was historically the motivation for the development of the theory of ellipticcurves.

Evidently, therefore, to give a meromorphic function on C=ƒ is to give a meromorphic functionon C which is invariant under ƒ, i.e., some doubly periodic function f W C ! C with periods!1 and !2; equivalently, some function such that f .z C m!1 C n!2/ D f .z/ D f .z C �/ for allz 2 C; m; n 2 Z; � 2 ƒ. One such function is the Weierstrass }-function associated to the latticeƒ, defined

}.z/ D }.zIƒ/ D1

z2C

X�2ƒnf0g

�1

.z � �/2�1

�2

�:

It may be shown ([Apo12], pp. 10–11.) that }.z/ is invariant under ƒ and analytic apart from adouble pole at each lattice point. Moreover, the function C=ƒ! P2.C/ given by

Œz� 7! .}.z/ W }0.z/ W 1/;

where Œz� D zCƒ, defines an embedding of the Riemann surface C=ƒ into the projective plane. Theimage of this embedding is a nonsingular projective plane curve defined over C by the projectiveequation

Y 2Z D 4X3 � g2XZ2� g3Z

3;

with

g2 D 60X

�2ƒnf0g

1

�4and g3 D 140

X�2ƒnf0g

1

�6:

The choice of notation is explained in appendix B.

Such a curve is an example of an elliptic curve, which is ([Mil06], p. 45) a nonsingular projectiveplane curve E of degree 3 over C together with a distinguished point O 2 E.C/. One often takes

51

the distinguished point to be the “point at infinity,” which is the image of 0 under the aboveisomorphism.

Two elliptic curves E and E 0 are isomorphic if there is an invertible change of variables transformingthe equation defining E into the equation defining E 0. It may be shown (op. cit., pp. 46–47) thata full set of isomorphism class representatives is given by the curves

E D E.a; b/ W Y 2Z D X3 C aXZ2 C bZ3; 4a3 C 27b2 ¤ 0;

with distinguished points O D .0 W 1 W 0/.Returning to the context of lattices: given that, to an arbitrary lattice ƒ we may associate anelliptic curve E.ƒ/ WD C=ƒ, we might reasonably ask when ƒ ¤ ƒ0 give rise to isomorphic ellipticcurves E.ƒ/ � E.ƒ0/.

If ƒ0 D cƒ for some c 2 C�, then ƒ and ƒ0 are said to be homothetic, and Œz� 7! Œcz� gives an

isomorphism C=ƒ0�! C=ƒ. In fact, the converse is true; that is,

E.ƒ/ � E.ƒ0/ if and only if ƒ D cƒ0 for some c 2 C:

As it clearly suffices to work with homothety classes of lattices, we may scale our lattice so that oneof the generators is 1, and the other (either !1

!2or !2

!1) lies in H; let us call this (non-real) generator

� , and choose the homothety class representatives

ƒ.�/ D 1 � ZC � � Z;

for any � 2 H. It may be shown ([Mil90], p. 41) that ƒ.�/ � ƒ.� 0/ if and only if there exists 2 �.1/ such that � 0 D � � ; that is, ƒ.�/ � ƒ. � �/ for any 2 �.1/.

Thus the map � 7! E.�/ WD E.ƒ.�// naturally induces an injection from �.1/nH to the space ofelliptic curves over C, modulo isomorphism; that is, �.1/nH serves as a parameter space for ellipticcurves over C. Consequently, the value of the j -invariant (appendix C, below) determines theisomorphism class of the elliptic curve.

52

Appendix B

Periods

We are familiar with the natural inclusions N � Z � Q � Q � C; however, there is an intermediateclass of numbers P;Q � P � C, which we define and investigate now. The primary reference is[KZ01], pp. 2–14.

A complex number z D x C iy is called a period if there exist domains Dx;Dy in Rn defined bypolynomial inequalities, and rational functions px.x/; py.x/ of x 2 Rn with rational coefficients,such that

x D

ZDx

px.x/dx and y D

ZDy

py.x/dx;

where the convergence of both integrals is absolute. It is immediate from this definition that theset of periods is countable. We also frequently consider the set bP WD P

�1�

�of extended periods.

Evidently every algebraic number is a period, as are certain transcendental numbers; for instance,

� D

Zx2Cy2�1

dx dy;

and similarly log x is a period for every rational x > 1, as

log x D

Z x

1

dt

t:

Furthermore, given an ellipse with rational major and minor radii a and b, one may compute itsperimeter to be a period:

I D

Z Zx2

a2Cy2

b2D1

dy dx D 2

Z b

�b

s1C

a2x2

b4 � b2x2dx:

This second expression in particular is an example of a class of integrals known as elliptic integrals;another example is given by the integral

p.y/ D p.yIg2; g3/ D

Z 1y

dtp4t3 � g2t � g3

;

53

where g2 and g3 are constants. The choice of notation is deliberate: if g2 and g3 are the constantsassociated to the Weierstrass elliptic function }.z/ D }.zIƒ/ from appendix A, then p.y/ is inverseto }. That is,

z D

Z 1}.z/

dtp4t3 � g2t � g3

:

If these constants and y are all algebraic, then p.y/ is a period ([Apo12], pp. 9–11 and 23).

To a given elliptic curve E we may associate two period integrals over E.C/, by integrating theholomorphic 1-form dx

yagainst fixed basis elements of H1

�E.C/IZ

�Š Z2. In particular, given the

elliptic curve Et defined by the projective equation Y 2Z D XZ.X � Z/.X � tZ/ for some t 2 C(the so-called Legendre equation), we have the period integrals

�1.t/ D

Z 1

t

dxpx.x � 1/.x � t /

and �2.t/ D

Z 11

dxpx.x � 1/.x � t /

;

which also arise as the solutions of the Picard-Fuchs equation

t .t � 1/�00.t/C .2t � 1/�0.t/C1

4�.t/ D 0:

We will return to these periods in the next appendix.

54

Appendix C

The period of a cusp form

There is a natural way to associate a period to a modular form. It is a fact ([Mil90], p. 12) that,if f .z/ is a modular form of weight 2 and t .z/ a modular function on H for some � � �.1/, thenthe multi-valued function F.z/ defined implicitly by F

�t .z/

�D f .z/ satisfies a linear third-order

differential equation with coefficients in Q. This suggests a connection with the period integralsencountered as the solutions of such equations in appendix B.

One classical example of a cusp form is the discriminant function, defined

�.z/ D q

1YnD1

.1 � qn/24; q D e2�iz :

This is a cusp form of weight 12 for the full modular group ([KZ01], pp. 13–14), and satisfies

�.z/ D

1XnD1

�.n/qn;

where �.n/ is the Ramanujan tau function. Certain properties of this function (for instance,its multiplicativity) were conjectured by Ramanujan, and it has been the object of ongoing studythroughout the 20th century; we do not explore this here.

A classical example of a modular function is the j -invariant (not to be confused with the factorof automorphy j. ; z/): for any z 2 H, define

j.z/ D1728g2.z/

3

�.z/; where g2.z/ D 60

X.m;n/2Z2

0 1

.mz C n/4;

the prime over the summation indicating as usual that the possibility m D n D 0 is excluded, andg2 the function from appendix A. The j -invariant has the additional property that it descends to

a bijection G.Z/nH��! C, and that the field of modular functions for G.Z/ on H is precisely the

field C.j / of rational functions in j.z/.

Now: we define the modular function

�.z/ D 1 ��.z=2/2=3�.2z/1=3

�.z/:

55

As the quotient of two modular forms of weight 12, �.z/ is certainly a modular function for �.1/;we also have the relationship with the j -invariant given by

j.z/ D256

�1 � �.z/C �.z/2

�3�.z/2

�1 � �.z/

�2 :

Now, recall the period integrals �1.t/ and �2.t/ from appendix B. These numbers are linearlyindependent over R ([Apo12], pp. 11–13), and so taking t D �.z/ we may use them to associate alattice

L.�/ WD �1��.�/

�� Z˚�2

��.�/

�� Z

to a given � 2 H (or, rather, a given � 2 �.1/nH). This lattice is homothetic to the latticeƒ.�/ D 1 � Z˚ � � Z; that is, we have either

� D�1

�2

��.�/

�or � D

�2

�1

��.�/

�:

As such, we see the relationship between the period � of the lattice ZC �Z and elliptic integrals. Inparticular, it suffices (by rescaling � if necessary) to assume �1

��.�/

�D 1, leaving only the integral

�2 under consideration.

Finally, we have seen that cusp forms f .z/ for � give rise to �-invariant differentials f .z/ dz onH�. Pairing these differentials (i.e., cohomology class representatives) against the cycle giving riseto �2.t/ yields the convergent integral Z i1

1

f .z/dz;

and because for any cusp form of weight 2k for �.1/ we haveZ 1

0

f .z/dz D a0.f / D 0

(recalling the constant term function a0.f / from section 2.1) we are ultimately led to the integralwe consider in this document, namely, Z i1

0

f .z/dz:

More generally: given a cusp form f .z/ of weight 2k, we may define the period polynomial ofdegree 2k � 2 by

rf .X/ D

Z i1

0

f .�/.� �X/2k�2 d�

(definition from [Zag91], eq. (1)). These period polynomials are of independent interest for severalreasons which are beyond the scope of this thesis; we direct the interested reader to a more thoroughtreatment in [Zag91] and [VC99].

One might ask why we pursue only the integral from 0 to i1, given the many other rational points˛; ˇ 2 Q between which we might want to integrate f .z/ dz; it is because the value of the periodwill allow us to compute the value of any integralZ ˇ

˛

f .z/dz; ˛; ˇ 2 Q [ fi1g:

56

Indeed ([Gun14], pp. 16–18), recall the Farey tessellation from section 1.1, and call a geodesicf˛; ˇg unimodular if ˛ and ˇ are Farey neighbours, i.e. if it belongs to the Farey tessellation, andis therefore a G.Z/-translate of the boundary of F , itself the union of three G.Z/-translates off0; i1g. We remark here also that, of course, 1 and i1 are identified in P1.Q/ as in H.

Now, suppose f˛; ˇg is a geodesic joining elements of Q [ fi1g, and without loss of generality letus assume f˛; ˇg D f0; p

qg (of course, if ˇ D i1 then f0; i1g is unimodular). Let us form the

continued fraction for pq

:

p

qD a0 C

1

a1 C1

a2 C1

a3 C1

: : : C1

ar

D ha0I a1; a2; : : : ; ari; a0 2 Z; ai 2 N for i 2 N;

and consider the convergentspk

qkWD ha0I a1; : : : ; aki:

Then it may be shown (loc. cit.) that fpk

qk;pkC1

qkC1g is unimodular. Consequently,

f0; pqg D f0; i1gC fi1; p1

q1g C f

p1

q1; p2

q2g C � � � C f

pr�1

qr�1; pr

qrg;

with each geodesic on the right-hand sideG.Z/-equivalent to f0; i1g. As we consider only subgroupsof finite index � in G.Z/, there will in general be only finitely many �-inequivalent geodesics in theFarey tessellation, and so this allows an effective way of computing all such integrals f˛; ˇg.

In the case of weight 2 modular forms, to which we restrict our attention, the period polynomialis a priori a constant. The dimensions of certain spaces M2.�/ and S2.�/ (for particular �) areknown ([SG07], chap. 6), and for many of these groups the space of cusp forms is null, while thespace of modular forms is not. That is, there are not enough cusp forms for the map f 7! rf tohave non-trivial image.

Thus we are led to try to define the notion of a period polynomial for an arbitrary modular form;however (at least in the case k D 1) the same formula

rf .X/ D

Z i1

0

f .�/ d� (C.1)

gives a divergent integral if f is not a cusp form, and so will not be helpful here. Furthermore, wewould like this to be a computable definition, in the sense that the period of a given modular formcan be explicitly evaluated. Such is the motivation for the classical period integral (1.4), which werecognize again as equation (C.1).

In [Ste12], Glenn Stevens solves the first problem by regularizing the integral (C.1) in a particularway. As for the second, we saw in sections 1.3 and 2.3 that it is enough to give a computabledefinition for the integral Z i1

0

E.z/ dz

when E.z/ is an Eisenstein series.

57

58

Appendix D

Supplementary calculations

In this appendix we prove various claims made elsewhere in the document, and investigate thevalues of the cocycles ��x

on arbitrary elements ˛ 2 G.Q/.

Lemma D.1. There is a right group action of G.Q/ on M2 given by the slash oper-ator; furthermore, this action permutes S2k.

Proof. It suffices to show that, if f is a modular form of level N , then f j˛ is a modular form oflevel N 0 for some N 0.

Suppose � is some congruence subgroup with �.N/ � � � �.1/, that f 2 M2.�/, and that˛ 2 G.Q/. Put �.˛/ WD ˛�1�˛ \ �; then if g D ˛�1g0˛ 2 �.˛/; g; g0 2 �, one has

.f j˛/jg D .f j˛/j.˛�1g0˛/ D f j.˛˛�1g0˛/ D f j.g0˛/ D .f jg0/j˛ D f j˛;

with the last equality holding because f jg0 D f for f 2M2.�/; g0 2 �. That is: if f is a modular

form for �, then f j˛ is a modular form for �.˛/.

Furthermore, we claim that �.˛/ is also a congruence subgroup: more precisely, if M 2 N is thesmallest positive integer such that M˛ is an integer matrix, we claim that �.˛/ � �.NM 2/. Forthis, it suffices to prove �.NM 2/ � ˛�1�.N/˛, the containment �.NM 2/ � �.N/ being clear.

Because of the Bruhat decomposition G.Q/ D B.Q/`B.Q/

��1

1

�B.Q/, where B � G is the

subgroup of upper-triangular matrices, and the further decomposition B D NA where N consists ofthe upper-unipotent matrices and A the diagonal matrices, it suffices to consider separately threeseparate conjugations.

1. In the first case, we have�1

�1

��1C aN bN

cN 1C dN

���1

1

�D

�1C dN �bN

�cN 1C aN

�:

Thus�1Ca0NM2 b0NM2

c0NM2 1Cd 0NM2

�clearly lies in the conjugate of �.N/ by

��1

1

�, for arbitrary

a0; b0; c0; d 0 2 Z.

59

2. In the case of an upper unipotent matrix with denominators at most M , it suffices to considerthe single matrix

�1 M�1

1

�. Thus�

1 � 1M

1

��1C aN bN

cN 1C dN

��1 1

M

1

�D

1C N

M.aM � c/ N

M2 .bM2 CM.a � d/ � c/

cN 1C NM.dM C c/

!:

Given

� D

�1C a0NM 2 b0NM 2

c0NM 2 1C d 0NM 2

�2 �.NM 2/;

we may take

a DM.c0 C a0M/; b D b0M 2C .d 0 � a0/M � c0; c D c0M 2; d D d 0M 2

� c0M;

to realize � as an element of�1 M�1

1

��1�.N/

�1 M�1

1

�.

3. Finally we conjugate by a diagonal element: as�1M

M

��1C aN bN

cN 1C dN

��M

1M

�D

�1C aN b N

M2

cNM 2 1C dN

�;

clearly arbitrary �1C a0NM 2 b0NM 2

c0NM 2 1C d 0NM 2

�2 �.NM 2/;

is written as such by taking

a D a0M 2; b D b0M 4; c D c0; d D d 0M 2:

Thus, if f is a modular form for �.N/, then f j˛ is a modular form for �.NM 2/. As such, theslash operator acts on the space M2 of weight 2 modular forms of all levels. Because the action of�.1/ permutes the cusps of H, we see that the action of the slash operator is well-defined on thesubspace of cusp forms.

Lemma D.2. The action of SL2.R/ on H by linear fractional transformations co-incides with left-multiplication under the identification of H with SL2.R/=SO2.R/.That is, if

z 2 H !�y1=2 xy�1=2

y�1=2

�K1 2 SL2.R/=K1;

where K1 D SO2.R/, then

az C b

cz C d2 H !

�a b

c d

��y1=2 xy�1=2

y�1=2

�K1

whenever ad � bc D 1.

60

Proof. One may write for any matrix in SL2.R/�a b

c d

�D

1p

c2Cd2

acCbdpc2Cd2

pc2 C d2

! dp

c2Cd2

�cpc2Cd2

cpc2Cd2

dpc2Cd2

!;

with the matrix on the right evidently in K1. In particular, one has�a b

c d

��y1=2 xy�1=2

y�1=2

�D

�ay1=2 .ax C b/y�1=2

cy1=2 .cd C d/y�1=2

�D

0@ 1pc2yC.cxCd/2y�1

acyC.axCb/.cxCd/y�1

pc2yC.cxCd/2y�1p

c2y C .cx C d/2y�1

1A dpc2Cd2

�cpc2Cd2

cpc2Cd2

dpc2Cd2

!

D

0@ y1=2

p.cy/2C.cxCd/2

acy2C.axCb/.cxCd/p.cy/2C.cxCd/2

y�1=2p.cy/2 C .cx C d/2

1A dpc2Cd2

�cpc2Cd2

cpc2Cd2

dpc2Cd2

!;

and because jcz C d j D jcx C d C icyj D .cy/2 C .cx C d/2, we see�a b

c d

��y1=2 xy�1=2

y�1=2

�K1 D

y1=2

jczCd jacy2C.axCb/.cxCd/

jczCd j

y�1=2jcz C d j

!K1:

We now compute

az C b

cz C dDaz C b

cz C d�c Nz C d

c Nz C dD.ax C b/.cx C d/C acy2

jcz C d j2C i

y

jcz C d j2;

and the claim is proven.

We will now calculate the values of ��xon arbitrary elements ˛ 2 G.Q/. First of all, let S D

.Q=Z/2; S 0 D S n f0g, and let

E �2 .Q/ D E2.Q/˚Q � �0:

Then the rule x 7! �x extends by linearity to give a surjective, Q-linear map

� W QŒS� �! E �2 .Q/;

sending QŒS 0� to E2.Q/. For any ˛ D�a bc d

�2 G.Z/, it may be shown ([Ste12], p. 58) that

�x DX

y2.Q=Z/2y˛Dx

�y j˛;

the sum taken over all vectors y 2 .Q=Z/2 such that

�y1 y2

� �a b

c d

�D�x1 x2

�:

61

One may use this identity to give an action of G.A/ on E �2 .Q/ ([Ste12], pp. 59–60): let K1 acttrivially and Kfin D

Qp G.Zp/ act componentwise on

.Q=Z/2 Dap

.Qp=Zp/2;

the identification obtained via the canonical isomorphism given by the first isomorphism theorem:

ker

Qp �������! Z

�1p

�=ZP1

nD�m anpn 7�!

P�1nD�m anp

n

!D Zp H) Qp=Zp Š ZŒ 1

p�=Z:

This defines an action of the subgroup K D K1 �Kfin � G.A/ on QŒS� which preserves the kernelof �. Thus we have a homomorphism

� W K �! AutQ.E�2 .Q//:

The theorem of strong approximation (theorem 3.1.2) states that G.A/ D G.Q/ � K, and if u; v 2K; ˛; ˇ 2 G.Q/ satisfy ˛u D vˇ, then our identity above shows

.�xj˛/j�.u/ D .�xj�.v//jˇ

for any x 2 S . Thus � extends to a homomorphism G.A/ ! AutQ.E�2 .Q//, extending the usual

action of G.Q/ on E �2 .Q/.

Let ! D

��1

1

�2 G.Q/ as before, and let �.!/ 2 Aut.E2/ with � the homomorphism above. Let

�! 2 Z1.G.Q/IHomC.E2;C// be the cocycle defined (for E 2 E2; ˛ 2 G.Q/) by

�!E .˛/ D ��E j�.!/.��1˛!/:

We now put

� D1

2.� C �!/ and � 0 D

1

2i.� � �!/

so that� D � C i� 0:

We will compute the “real” and “imaginary” parts � and � 0 separately.

First of all: for E 2 E2, let e!E W G.Q/! C be defined

e!E .˛/ D �eE j�.!/.!�1˛!/:

Define the Dedekind symbol sE W P1.Q/! C to be

sE .r/ D

(12

�eE�1 r1

�C e!E

�1 r1

��if r 2 Q;

0 if r D i1:

We also define the complementary function s0E W P1.Q/! C by putting s0E .i1/ D 0 and for r 2 Q

implicitly by the equatuioneE�1 r1

�D sE .r/C i � s

0E .r/:

62

By proposition theorem 2.2.6 (b), we know for ˛ D�a bc d

�2 G.Q/ with c � 0, that

�E .˛/ D

(aca0.E/C

dca0.Ej˛/ if c > 0;

bda0.E/ if c D 0;

and� 0E .˛/ D �s

0E

�ac

�:

By the discussion of section 2.4, the functions �x span E2, and so to complete our calculations weneed only compute the functions s�x

; s0�x.

We close with two results that allow us to explicitly compute �f .˛/ for arbitrary ˛ 2 G.Q/.

Lemma D.3. Let x D .x1; x2/ 2 S0, and let m; n 2 Z with n > 0 and .m; n/ D 1, so

that mn2 P1.Q/ n f0g.

(a) One has

s�x

�mn

�D

n�1X�D0

B1�x1C�n

�B1�m � x1C�

nC x2

�:

(b) Let F W S 0 ! C be defined

F.x/ D �ıx10 log j1 � q.x2/jI

then

s0�x

�mn

�D

1

2�

�F.x / � F.x/

�;

where 2 G.Q/ has � i1D mn

.

Proof. [Ste12], pp. 72–73.

Summarizing, we obtain

Lemma D.4. Let x 2�Q=Z

�2nf0g and let ˛ D

�a b

c d

�2 G.Q/ with a; b; c; d all

pairwise coprime and c � 0. Then ��xD ��x

C i� 0�x, where

��x.˛/ D

(aca0.�x/C

dca0.�xj˛/ if c > 0;

bda0.�x/ if c D 0;

and

� 0�x.˛/ D

(12�

�F.x / � F.x/

�if c > 0;

0 if c D 0;

where 2 G.Q/ has � i1D ac

and F.x/ D �ıx10 log j1 � q.x2/j.

In particular,

Corollary 1. Let � be a congruence subgroup and let E 2 E2.�/. Then � 0E . / D 0

and so �E . / D �E . /.

63

64

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