Computational methods forBelyi maps and dessins

John VoightDartmouth College

The Impact of Computation on Number TheoryNational Center for Theoretical Sciences (NCTS)

Hsinchu, Taiwan2 August 2014

Computers in Number Theory

Atlas Symposium No. 2 on Computers in Number Theory was heldat Oxford during the week of 18–23 August 1969.

Computers in Number Theory: preface

xii PREFACE

About 60 invitations sent out were not accepted for a variety of reasons, mainly those of previous commitment. On Wednesday evening there was an open discussion on "Number Theoretic Subroutines and Tables". As might have been expected with so large a number of original thinkers no substantial agreement was reached, but the three following conclusions commanded the support of most of the 40 people present. First, that a complete list of what exists, in print and privately, would be useful. Second, that the subject is too various to lend itself to an agreed package of subroutines and a uni­versal language. Third, that some facility should exist for making known the "results" of unsuccessful but substantial computation. The current tradition of mathematics does not look kindly on a statement in print that sulphuric acid was poured on a white powder without apparent reaction, but machine time can be expensive. A tape recording of the discussion is kept at the Atlas Computer Laboratory.

There were also some distractions not directly number theoretic. On Monday evening Professor Mordell gave us a fascinating account of his in­finitely varied mathematical and personal experience, entitled "Reminiscences of an Octogenarian Mathematician". On Tuesday Dr. Howlett's sherry party at the Queen's College made serious thought impossible after 6 p.m. On Wednesday afternoon, there was an organized tour of the Atlas Laboratory in action. On Friday, the Symposium dinner at the Queen's College provided a memorable conclusion to the proceedings.

A large number of people and public bodies made the Symposium both possible and successful. First and foremost, Jack Howlett, without whose forethought and determination there would have been no background on which to build. The Science Research Council, through the Atlas and Ruther­ford Laboratories, provided the secretarial staff, and gave a substantial sum to cover the expenses of some of the speakers. International Computers Limited, the manufacturers of the Atlas computer, also contributed gener­ously towards these expenses. Synolda Butler did all the hard secretarial work involved in planning and organizing the Symposium. The Programme Committee, consisting of the editors, R. F. Churchhouse, and H. P. F. Swinnerton-Dyer, had only to sign her letters, and we became increasingly aware of her exact attention to every detail.

At Oxford, most of the participants were housed in Jesus College, whose authorities gave us every cooperation. Their rooms were not big enough to accommodate the numbers at the sherry party and the dinner, and we were fortunate in our choice of the Queen's College for these occasions. The actual meetings were held at the Mathematics Institute, by kind permission of Professor Higman; he and the Institute staff were very helpful. The arrange­ments for travel were in the capable hands of "Robbie", Mr. C. L. Roberts of the Atlas Laboratory. He also attended to the important and easily for-

PREFACE xiii

gotten details that are nobody's business in particular, dealing imperturbably with all emergencies.

Finally we must thank the authors for their cooperation, and Academic Press, particularly the production staff, for their frequent and ready advice during publication. The editors have confined themselves mainly to making small alterations of style or notation. The articles are either the original manu­scripts given to us at the time of the Symposium or revised versions received after it; they are printed more or less in the same order as the talks were given, since we soon found that many of them would not fit in to any rigid scheme of classification.

May, 1971

A. 0. L. ATKIN B. J. BIRCH

Computers in Number Theory: published elsewhere

Editors' Note Several of the papers presented at the Symposium are published elsewhere.

Atkin, A. 0. L. (Atlas Laboratory) and Swinnerton-Dyer, H. P. F. (Cambridge) (1971).

Modular forms on non-congruence subgroups. Jn "Proceedings of Syn1posia in Pure Mathematics, XIII". American Mathematical Society, Providence. Berlekamp, E. R. (Bell Laboratories) (1970). Factoring polynomials over large finite fields. Math. Comp. 24, 713-735. Birch, B. J. (Oxford) (1969). K2 of global fields. In "Proceedings of Symposia in Pure Mathematics, XX". Number Theory Institute. American Mathematical Society, Providence. Froberg, C.-E. (Lund) (1968). On the prime zeta function. BIT 8, 187-202.

Frohlich, A. (King's College London) (1969). On the classgroup of integral grouprings of finite abelian groups. Mathematika 16, 143-152.

Good, I. J. and Churchhouse, R. F. (1968). The Riemann hypothesis and pseudorandom features of the MObius sequence. Math. Comp. 22, 857-862.

Hasse, H. (Hamburg) and Liang, J. (1969). Ober den Klassenk:Orper zum quadratischen ZahlkOrper mit der Diskriminante -47, Acta Arithmetica 16, 89-98.

Riesel, H. (Stockholm) (1969). Lucasian criteria for the primality of N=h.2"-1. Math. Comp. 23, 869-875.

ix

Atkin–Swinnerton-Dyer

MODULAR FORMS ON NONCONGRUENCE SUBGROUPS

A. 0. L. ATKIN AND H. P. F. SWTNNERTON-DYER

1.1. Introduction. This paper is an interim account of work still in progress. It started with an intention to discover something about modular forms on non­congruence subgroups of the classical modular group. We were successively led into function theory, group theory, computing, numerical analysis, and combinatorics. This last gives rise to our rather slender justification for including the paper in this volume, though we should say that at the time when one of us was asked to speak at the Symposium, the combinatorial aspect seemed relatively greater than it does now.

The main outcome of our work is the discovery of objects associated with forms on modular subgroups which, in the congruence case, are just the eigenvalues of the Heeke operators T". By ••discovery'' we imply total confidence in their existence, without at present any foreseeable prospect of proof. We are not yet even in a position to make formal conjectures, since there remain some marginal uncertainties as to what properties these objects enjoy. We hope however that the reader who gets as far as §5 will share our certainty of their existence.

Apart from the interest of this discovery, we think that there is some general interest in the techniques by which we arrived at it. From the beginning, we ap­proached the problem with the possibility of using computers in mind. In the event, much of what we have done would have been impossible without their aid, and even those who deny the validity of a proof requiring computed facts cannot cavil at conjectures arrived at by inspection of them. We consider that there must remain large areas of mathematics, and of number theory in particular, where computer­aided investigations should produce significant results. For this reason, we have attempted to make this account generally intelligible, in some places preferring simplicity and explicit example to stating the most general theorem possible.

We thank the Directors of the Atlas Computer laboratory and the Cambridge University Mathematical Laboratory for making the necessary machine time avail­able, and Mr. Stephen Muir of the Atlas Computer Laboratory for his assistance with some of the programming.

1.2. The modular group. The classical modular group r consists of all linear fractional transformations V-r=(a-r+b)/(c-r+d), where a, b, c, and dare rational

Copyright C 1971, American Mathematical Society

Belyı maps and permutation triples

A Belyı map is a nonconstant morphism φ : X → P1 of curvesover C unramified away from 0, 1,∞.

Belyı proved that a curve X over C can be defined over Q if andonly if X admits a Belyı map. Grothendieck said of this result:“never, without a doubt, was such a deep and disconcerting resultproved in so few lines!”

There is a bijection between Belyı maps of degree d ≥ 1 up toisomorphism and transitive permutation triples of degree d ,

σ = (σ0, σ1, σ∞) ∈ S3d such that σ∞σ1σ0 = 1 and

〈σ0, σ1, σ∞〉 ≤ Sd is a transitive subgroup

up to simultaneous conjugation in Sd . If σ corresponds to φ, wesay that φ has monodromy σ.

Combinatorial aspects

The cycles of the permutation σ0 (resp. σ1, σ∞) correspond to thepoints of X above 0 (resp. 1,∞) and the length of the cyclecorresponds to its multiplicity. The genus of a Belyı map is givenby the Riemann–Hurwitz formula:

g = 1− d +e(σ0) + e(σ1) + e(σ∞)

2

where e(σ) is the excess of σ, equal to n minus the number ofdisjoint cycles in σ.

For example, the permutation triple

σ0 = (1 2)(3 4)(5 6), σ1 = (1 4)(2 5)(3 6), σ∞ = (1 3 5)(2 6 4)

corresponds to a Belyı map of genus 1− 6 + (3 + 3 + 4)/2 = 0f : P1 → P1:

φ(x) =27x2(x − 1)2

4(x2 − x + 1)3and φ(x)−1 = −(x − 2)2(2x − 1)2(x + 1)2

4(x2 − x + 1)3.

DessinsIn Esquisse d’un Programme, Grothendieck associates to a Belyımap φ : X → P1 the dessin φ−1([0, 1]) ⊂ X , a bicolored graphembedded on the surface X , with vertices above 0 and above 1labelled white and black, respectively.

For example, for the triple

σ0 = (1 2 3), σ1 = (1 2)(3 4), σ∞ = (2 3 4)

we have

f (x) = −(x − 1)3(x − 9)

64x= 1− (x2 − 6x − 3)2

64x

and dessin

3− 2√

3

1

3 + 2√

3

9

A computational Esquisse

Grothendieck asks:

Exactly which are the conjugates of a given orientedmap? I considered some concrete cases (for coverings oflow degree) by various methods... I doubt that there is auniform method for solving the problem by computer.

Is there an algorithm that takes

input: a permutation triple

and produces

output: a model for the corresponding Belyı map over Qthat runs in time doubly exponential in the degree d?

Galois acting on dessins

The Galois action on dessins is mysterious and highlyunpredictable, and nontrivial invariants are difficult to find.

(Drawn by Frits Beukers)

Applications

I Action of Gal(Q/Q) on dessins (Grothendieck);

I Covering curves with many automorphisms (Wolfart);

I Shimura curves, coding theory (Elkies);

I Algebraic solutions of differential equations (Vidunas–vanHoeij);

I ABC polynomials (Beukers, Watkins) and specializations75319694514582 − 3842427663 = 14668;

I Physics and moonshine;

I Inverse Galois theory (Malle–Matzat, . . . );

I Number fields with small ramification set (Jones–Roberts,Malle);

I ...

Computing Belyı maps

Hundreds of papers have been written on methods to computeBelyı maps: these are surveyed in joint work with Jeroen Sijsling.

There are four general techniques to compute Belyı maps:

1. Direct methods, where a system of equations is solved viaGrobner basis;

2. Complex analytic methods, where iterative numerical methodsare employed and the exact solution is recognized;

3. p-adic methods, where a solution is found modulo p, liftedand recognized analogously; or

4. modular forms methods, where one computes intrinsically onquotients of the upper half plane.

Direct method: example

Consider the triple

σ0 = (1 6)(2 5)(3 4), σ1 = (0∞ 1)(2 4 6), σ∞ = (0 1 4 3 2 5 6∞)

(coming from G = PGL2(F7) acting on P1(F7)). The Belyı mapf : X → P1 has X of genus 0. Taking the totally ramified point at∞, the map f is given by a polynomial j(t) ∈ Q[t] such that

j(t) = a(t)2b(t) and j(t)− c = d(t)3e(t)

where deg a = 3 and deg b = deg d = deg e = 2 and c ∈ Q.

Direct method: equations

Equating coefficients gives an ideal generated by 10 equations in10 unknowns:

a20b0c − cd3

0 e0,

2a1a0b0c + a20b1c − 3cd1d

20 e0 − cd3

0 e1,

2a2a0b0c + a21b0c + 2a1a0b1c + a2

0c − 3cd21d0e0 − 3cd1d

20 e1 − cd3

0 − 3cd20 e0,

2a2a1b0c + 2a2a0b1c + a21b1c + 2a1a0c + 2a0b0c − cd3

1 e0 − 3cd21d0e1 − 3cd1d

20

− 6cd1d0e0 − 3cd20 e1,

a22b0c + 2a2a1b1c + 2a2a0c + a2

1c + 2a1b0c + 2a0b1c − cd31 e1 − 3cd2

1d0 − 3cd21 e0

− 6cd1d0e1 − 3cd20 − 3cd0e0,

a22b1c + 2a2a1c + 2a2b0c + 2a1b1c + 2a0c − cd3

1 − 3cd21 e1 − 6cd1d0 − 3cd1e0 − 3cd0e1,

a22c + 2a2b1c + 2a1c + b0c − 3cd2

1 − 3cd1e1 − 3cd0 − ce0,

2a2c + b1c − 3cd1 − ce1.

Direct method: solution

Solving via a Grobner basis, we find

j(t) =(

2√

2t3 − 2(2√

2 + 1)t2 + (−4 + 7√

2)t + 1)2

·(

14t2 + 6(√

2 + 4)t − 8√

2 + 31).

with

j(t)− 432(4√

2− 5) =(2t2 − 2

√2 + 1

)3(14t2 − 8(

√2 + 4)t − 14

√2 + 63

).

There is also a differentiation trick due to Atkin–Swinnerton-Dyerthat partially saturates this ideal. Nevertheless, because of thecomplexity of Grobner basis techniques, the direct method onlyworks in a small range of examples.

Our main result

Our main result: we provide a general-purpose numerical methodfor the computation of Belyı maps. Not a rigorous algorithm (yet),no estimate of the running time, but we can verify that output iscorrect.

This is joint work with Michael Klug, Michael Musty, and SamSchiavone.

Our method uses modular forms (fourth type). Two mainingredients:

(a) algorithms for finite index subgroups of triangle groups, and

(b) power series expansions.

Triangle groups

To a, b, c ∈ Z≥2, we associate the triangle group

∆(a, b, c) = 〈δa, δb, δc | δaa = δbb = δcc = δcδbδa = 1〉

and the quantity χ(a, b, c) = 1/a + 1/b + 1/c − 1 ∈ Q. Thetriangle group ∆(a, b, c) is the orientation-preserving subgroup ofthe group generated by the reflections in the sides of a trianglewith angles π/a, π/b, π/c in H.

za zc

zb

−zc

b

b

bb

b

b

bb

τa

τb

za zc

zb

−zc

δa

δb

b

b

bb

b

b

b b

In essentially all cases, χ(a, b, c) < 0, and H = H is the upperhalf-plane; we say (a, b, c) is hyperbolic.

Triangle groups and permutation representations

Associated to a transitive permutation triple σ is a homomorphism

∆(a, b, c)→ Sd

δa, δb, δc 7→ σ0, σ1, σ∞

where a, b, c are the orders of σ0, σ1, σ∞, respectively. Thestabilizer of a point Γ ≤ ∆(a, b, c) has index d , and the abovehomomorphism is recovered by the action of ∆ on the cosets of Γ.The quotient map

φ : X = Γ\H → ∆\H

then realizes the Belyı map with monodromy σ, so from thisdescription we have a way of constructing the Belyı map associatedto σ.

Coset graph, fundamental domain, and dessin

σ0 = (1 5 4 3 2), σ1 = (1 6 4 2 3 5), σ∞ = (1 4 3 6)

1

δa

δ2a

δ−2

a

δ−1

a

δb

δaδa

δa

δa

δa

δb

δb

δb

δb δb

δb

δa

s1

s1

s2

s2

s3

s3

s4

s4 s5

s5

s6

s6

bc b

×

bc

b×

bc

b

× bc

b

×

bc

b

×

bc

b

×

1

2

3

4

5

6

Algorithms for triangle groups

Theorem

There exists an algorithm that, on input a hyperbolic transitivepermutation triple σ ∈ S3

d , gives as output:

I A coset graph for Γ ≤ ∆(a, b, c) associated to σ;

I A fundamental domain DΓ for Γ with side pairing elements;

I The associated dessin drawn conformally correct on DΓ;

I A reduction algorithm for Γ; and

I A presentation for Γ with a solution to the word problem in Γ.

Power series expansions

Now we compute equations for the Belyı map φ numerically.

The main algorithmic tool for this purpose is the method of powerseries expansions (joint work with John Willis).

Our method is inspired by the method of Stark and Hejhal, whoused the same basic principle to compute Fourier expansions forMaass forms on SL2(Z) and the Hecke triangle groups ∆(2, p,∞).

Modular forms, no cusps!

A modular form f of weight k ∈ 2Z≥0 for a cocompact Fuchsiangroup Γ is a holomorphic map f : H → C satisfying

f (γz) = (cz + d)k f (z)

for all γ =

(a bc d

)∈ Γ.

As Γ is cocompact, the quotient X = Γ\H has no cusps, so thereare no q-expansions!

However, not all is lost: such a modular form f still admits apower series expansion in the neighborhood of a point p ∈ H.

Power series expansions in unit disc

A q-expansion is really just a power series expansion at ∞ in theparameter q, convergent for |q| < 1. So it is natural to consider aneighborhood of p normalized so the expansion also converges inthe unit disc D for a parameter w . So we map

w : H → D

z 7→ w(z) =z − p

z − p.

We then consider series expansions of the form

f (z) = (1− w)k∞∑n=0

bnwn

where w = w(z).

Example

Let f ∈ S2(Γ0(11)) be defined by

f (z) = q∞∏n=1

(1− qn)2(1− q11n)2 = q − 2q2 − q3 + 2q4 + ....

Consider expansions about the point p = (−9 +√−7)/22 ∈ H, a

CM point on X0(11) for K = Q(√−7). From the q-expansion:

f (z) = −√

3 + 4√−7Ω2(1− w)2·

·(

1 + Θw +5

2!(Θw)2 − 123

3!(Θw)3 − 59

4!(Θw)4 − . . .

)where

Θ =−4 + 2

√−7

11πΩ2

and Ω = 0.500491 . . . is the Chowla-Selberg period for K .

Basic idea

w

gw

Basic idea

Let D ⊂ D be a fundamental domain for Γ contained in a circle ofradius ρ > 0. Let f ∈ Sk(Γ). We consider an approximation

f (z) ≈ fN(z) = (1− w)kN∑

n=0

bnwn

valid for all |w | ≤ ρ to some precision ε > 0.

For a point w = w(z) 6∈ D, there exists g ∈ Γ such thatw ′ = gw ∈ D; by the modularity of f we have

fN(z ′) ≈ f (z ′) = j(g , z)k f (z)

(1− w ′)kN∑

n=0

bn(w ′)n ≈ j(g , z)k(1− w)kN∑

n=0

bnwn,

imposing a (nontrivial) linear relation on the unknowns bn.

Genus three example: setup

Consider the permutation triple σ = (σ0, σ1, σ∞), where

σ0 = (1 7 4 2 8 5 9 6 3)

σ1 = (1 4 6 2 5 7 9 3 8)

σ∞ = (1 9 2)(3 4 5)(6 7 8).

These permutations generate a transitive subgroup

G ∼= Z/3Z o Z/3Z ≤ S9

of order 81.

The corresponding group Γ ≤ ∆(9, 9, 3) = ∆ has index 9 hassignature (3;−). We compute the Belyı map

X (Γ) = Γ\H → X (∆) = ∆\H ∼= P1

Genus three example: fundamental domain and dessin

s1

s1

s2

s2

s3

s3

s4

s4

s5

s5

s6

s6

s7

s7s8

s8

s9

s9

bc b

×

bc

b ×

bc

b

×

bc

b

×

bc

b

× bc

b

×

bc

b

×

bc

b

×

bc

b

×

1

2

3

4

5

6

7

8

9

Label Coset Representative

1 12 δ3a3 δ−1

a

4 δ2a5 δ−4

a

6 δ−2a

7 δa8 δ4a9 δ−3

a

Label Side Pairing Element

s1 δbδ−2a

s2 δ−1b δ−4

a

s3 δaδbδ3a

s4 δaδ−1b δ4a

s5 δ−1a δbδ

−4a

s6 δ−1a δ−1

b δ3as7 δ2aδbδ

2a

s8 δ−2a δbδ

−3a

s9 δ3aδbδ4a

Genus three example: modular forms

The space S2(Γ) of cusp forms of weight 2 for Γ has dimensiondimC S2(Γ) = g = 3.

We find the echelonized basis

x(w)

(1− w)2= 1− 40

6!(Θw)6 +

3080

9!(Θw)9 − 1848000

12!(Θw)12 + . . .

y(w)

(1− w)2= (Θw) +

4

4!(Θw)4 +

280

7!(Θw)7 − 19880

10!(Θw)10 + . . .

z(w)

(1− w)2= (Θw)3 − 120

6!(Θw)6 − 10080

9!(Θw)9 − 2698080

12!(Θw)12 + . . .

where Θ = 1.73179 . . .+ 0.6303208 . . .√−1. The apparent

integrality of these coefficients are conjectured.

Genus three example: equations

We now compute the image of the canonical map

X (Γ) = Γ\H → P2

w 7→ (x(w) : y(w) : z(w));

we find a unique quartic relation

216x3z − 216xy3 + 36xz3 + 144y3z − 7z4 = 0

so at least numerically the curve X is nonhyperelliptic. Evaluatingthese power series at the ramification points, we find that theunique point above f = 0 is (1 : 0 : 0), the point above f = 1 is(1/6 : 0 : 1), and the three points above f =∞ are (0 : 1 : 0) and((−1± 3

√−3)/12 : 0 : 1).

Genus three example: equations

The uniformizing map f : X (Γ)→ X (∆) ∼= P1 is given by thereversion of an explicit ratio of hypergeometric functions:

f (w) = −1

8(Θw)9 − 11

1280(Θw)18 − 29543

66150400(Θw)27 + O(w36).

Using linear algebra, we find the expression for f in terms of x , y , z :

f (w) =−27z3

216x3 − 108x2z + 18xz2 − 28z3.

Having performed this numerical calculation, we then verify on thecurve X (Γ) that this rational function defines a three-point coverwith the above ramification points.

Final example

We compute a Belyı map f : P1 → P1 of degree 50, correspondingto the beautiful identity

26a(t)5b(t)− 510c(t)7 = d(t)2e(t)

where

a(t) = (t4 + 11t3 − 29t2 + 11t + 1)

· (64t5 − 100t4 + 150t3 − 25t2 + 5t + 1)

b(t) = 196t5 − 430t4 + 485t3 − 235t2 + 30t + 4

c(t) = t(t + 1)(2t2 − 3t + 2)(8t3 − 32t2 + 10t + 1)

d(t) = 28672t20 − 2114560t19 + · · · − 1520t2 − 240t − 8

e(t) = 16384t10 − 34960t9 − · · ·+ 705t2 + 110t + 4.

Plugging in t = 1, we get 52 · 195 + 618 · 1032 = 137.

Final example: fundamental domain

Final example

Write26a(t)5b(t)− 510c(t)7 = d(t)2e(t)

p(t)− q(t) = r(t).

The discriminant of p(t)− zq(t) is

disc(p(t)− zq(t)) =556071092

21918z36(z − 1)20;

the Galois group of f (t)− z over Q(√−7)(z) is

PSU3(F5) = g ∈ SL3(F25) : gσ(g)t = 1/scalars

where σ is the entry-wise (5th power) Frobenius. We have# PSU3(F5) = 126000 = 2432537 and PSU3(F5) → S50 via theaction on the subgroup A7 ≤ G of index 50.

Specializing at t = 2 yields a number field K of degree 50 ramifiedonly at 2, 5, 7 with Galois group PSU3(F5) :2.