Computing with modular forms and modular abelian varieties

U.C. Berkeley Ph.D. Dissertation

William A. Stein July 16, 1999

Preface

University of California, Berkeley William. A. Stein July, 1999

Contents

1 Explicitly constructing modular forms 4

1.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 The definition of modular symbols . . . . . . . . . . . . . 4 1.1.2 Modular symbols are dual to modular forms . . . . . . . . 5 1.1.3 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Method of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Arithmetic of quaternion algebras . . . . . . . . . . . . . . . . . . 8

2 Modular algorithms 9 2.1 Basis of eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Modular degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Period lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Rational parts of special values of L-functions . . . . . . . . . . . 9 2.6 Component groups . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Motivating problems 10

3.1 Visisibility of Shafarevich-Tate groups and the Birch and Swinnerton-

Dyer conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Modularity of a complex A_{5} Galois representation . . . . . . . . 10 3.3 Rational points on X_{0}(p) over Q(μ_{p}) . . . . . . . . . . . . . . . . 10 3.4 Serre’s conjecture modulo pq? . . . . . . . . . . . . . . . . . . . . 10

4 Tables 11

Introduction

The object of numerical computation is theoretical advance.

–A. O. L. Atkin

The main goal of this thesis is to present methods to explicitly compute with modular forms, Hecke operators, and the abelian varieties attached to them. The definition of the spaces S_{k}(Γ) of modular forms as functions on the upper half plane satisfying a certain equation is very abstract. The definition of the Hecke operators even more so. Nevertheless, one wants to carry out explicit numerical investigations on these objects.

We are fortunate that we now have methods available which allow us to transform the vector space of cusp forms of given weight (at least 2) and level N into a concrete object, which can be explicitely computed. We have the work of Atkin-Lehner, Birch-Swinnerton-Dyer, Cremona, Manin, Mazur, Merel, Tate, and many others to thank for this [BK75, Cre97, Mer94]. The Eichler-Selberg trace formula, as extended in [Hij74], can be used to compute characteristic polynomials of Hecke operators and hence gain some information about spaces of modular forms. Other methods include computing the Hecke operators and qexpansions using Brandt matrices and quaternion algebras as in [Koh98, Piz80, Piz76a, Piz76b, Piz76c], or the module of enhanced supersingular elliptic curves as exploited by Mestre-Oesterlé [Mes86]. Though each of the above methods are beautifully suited to certain applications, we follow the approach of Cremona’s book [Cre97] in viewing modular symbols as central.

Chapter 1 reviews the literature on modular curves, modular symbols, the method of graphs, and quaternion algebras. Chapter 2 then gives several algorithms, some possibly new. This thesis has as its central motivation several fascinating open problems. In Chapter 3 we present the results of our investigations into the following questions/conjectures:

• Visibility of Shafarevich-Tate groups and the Birch and Swinnerton-Dyer conjecture for analytic rank zero modular abelian varieties. (Joint with A. Agashe and B. Mazur.)

• The Artin conjecture on modularity of two dimensional complex Galois representations. (Joint with K. Buzzard.)

• Rational points on the modular curve X_{0}(p) over the field Q(μ_{p}). (Joint with L. Merel.)

• A possible extension of Serre’s conjectures to Galois representations modulo pq. (Joint with B. Mazur.)

Chapter 1

Explicitly constructing modular forms

Introduction to background.

1.1 Modular symbols

Modular symbols can be used to compute a Hecke module which is closely related to the space of modular forms of given weight k ≥ 2 for any congruence subgroup Γ of SL_{2}(Z). In this section we review the theory of Manin and Shokurov. Our presentation follows [Cre97, Mer94].

Fix a finite index subgroup Γ ⊂ SL_{2}(Z) and a positive integer k. If k is odd, assume that −1 ∉ Γ (otherwise the theory is empty).

1.1.1 The definition of modular symbols

Let M be the torsion free abelian group generated by symbols {α, β} with α, β ∈ P^{1}(Q) = Q ∪ {∞} subject to the following relations

{α, β} + {β, γ} + {γ, α} = 0, all α, β, γ ∈ P^{1}(Q).

Thus M can be obtained from the free abelian group on symbols {α, β} by quotienting out by the above 3-term relations and then by the torsion subgroup of the quotient. In particular, {α, β} = −{β, α} and {α, α} = 0. Next let

M_{k} = V_{k−2} ⊗ M

where V_{k−2} ⊂ Z[X, Y] is the subgroup of homogeneous polynomials of degree k − 2.

Define an action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Q) on P ∈ V_{k−2} by

gP (X, Y) = P (det(g)g^{−1} (\begin{matrix}X \\ Y \end{matrix}))
= P ((\begin{matrix}d & −b \\ −c & a \end{matrix}) (\begin{matrix}X \\ Y \end{matrix}))
= P (dX − bY, −cX + aY).

This is a left action because

(gh)P (v) = P (det(gh)(gh)^{−1}v)
= P (det(h)h^{−1} det(g)g^{−1}v)
= gP (det(h)h^{−1}v) = g(hP (v)).

Define an action of g on α ∈ P^{1}(Q) by fractional linear transformation:

gα = \frac{aα + b}{cα + d} ∈ P^{1}(Q).

Define an action of g on M_{k} by

g(P ⊗ {α, β}) = gP ⊗ {gα, gβ}

and extending linearly.

Let M_{k}(Γ) be the torsion-free quotient of M_{k} obtained by imposing the relations gx = x for all x ∈ M_{k} and all g ∈ Γ. Thus M_{k}(Γ) is the largest torsion free quotient of M_{k} on which Γ acts trivially:

M_{k}(Γ) := H_{0}(Γ, M_{k})/(torsion) (group homology).

We call the elements of M_{k}(Γ) modular symbols of weight k for Γ, and denote by P {α, β} the image of P ⊗ {α, β} in M_{k}(Γ).

1.1.2 Modular symbols are dual to modular forms

Theorem 1.1. There is a nondegenerate pairing

(M_{k}(Γ) ⊕ \overline{S}_{k}(Γ)) × M_{k}(Γ) ⟶^{⟨ , ⟩} C  (1.1)

given by

⟨f ⊕ g, P \{α, β\}⟩ = ∫_{α}^{β} f(z)P (z, 1)dz + ∫_{α}^{β} g(\overline{z})P (\overline{z}, 1)d\overline{z}

where the integral is along any geodesic path from α to β.

Proof. Merel – but for the Eisenstein part I’m guessing and better check this! I should check directly here that the pairing is well-defined.

1.1.3 Linear Operators

In this section we assume Γ = Γ_{0}(N) or Γ_{1}(N).

Hecke operators

Let Δ ⊂ GL_{2}(Q) be a subsect such that ΓΔΓ^{−1} = Δ and such that Γ\Δ is finite. Let R be a set of representatives of Γ\Δ, and for x ∈ M_{k}(Γ), define T_{Δ}(x) = ∑_{δ∈R} δx. We obtained a well-defined linear map

T_{Δ} : M_{k}(Γ) ⟶ M_{k}(Γ)

Assume that Γ is either Γ_{0}(N) or Γ_{1}(N). Let Δ_{n} ∈ M_{2}(Z) be the set of matrices of determinant n such that N | c and (N, a) = 1 (if Γ = Γ_{1}(N) impose the additional condition that N | (a − 1)). Define the nth Hecke operator T_{n} := T_{Δ_{n}}. When n = p is prime, the Hecke operator is thus

T_{p}(x) = [(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) + ∑_{r mod p} (\begin{matrix}1 & r \\ 0 & p \end{matrix})] x

(omit the first matrix if p | N).

Proposition 1.2. The pairing (1.1) respects the action of Hecke operators: ⟨fT_{n}, x⟩ = ⟨f, T_{n}x⟩ for all Hecke operators T_{n}, symbols x ∈ M_{k}(Γ), and modular forms f ∈ M_{k}(Γ) ⊕ \overline{S}_{k}(Γ).

The ∗-involution

The matrix J = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) defines an involution ∗ of M_{k}(Γ) given by x ↦ x^{∗} = Jx. More explicitely,

(P (X, Y){α, β})^{∗} = P (X, −Y){−α, −β}.  (1.2)

If f is a modular form, let f^{∗} be the holomorphic function \overline{f(−\overline{z})}, where the bar denotes complex conjugation. The Fourier coefficients of f^{∗} are the conjugates of those of f.

Proposition 1.3. The pairing (1.1) behaves as follows respects the action of ∗:

⟨x^{∗}, f^{∗}⟩ = \overline{⟨x, f⟩}.  (1.3)

Warning: Our choice of ∗ agrees with [Cre97] but not with [Mer94].

The Atkin-Lehner involutions

In this section, assume that the weight k is even. There is an involution w_{q} of M_{k}(Γ) associated to each prime power q exactly dividing N. Using the Euclidean algorithm, choose integers a and b so that aq − b(N/q) = 1. If we let

g = ( \begin{matrix}aq & b \\ N & q \end{matrix} ), then

w_{q}(x) = \frac{1}{N^{\frac{k}{2} −1}} · gx  (1.4)

We have w_{q}^{2} = 1 and if we let w_{q}(f) = f|_{w_{q}}, then ⟨w_{q}(f), x⟩ = ⟨f, w_{q}(x)⟩.

New and old symbols

[This section may contain errors.]

In this section, we assume Γ = Γ_{1}(N) to fix notation, though Γ_{0}(N) could be substituted just as well. Let M be a positive divisor of N, and let d be a divisor of N/M. We have well-defined linear maps

α_{d} : M_{k}(Γ_{1}(N)) → M_{k}(Γ_{1}(M))  α_{d}(x) = (\begin{matrix}d & 0 \\ 0 & 1 \end{matrix}) x (1.5)
β_{d} : M_{k}(Γ_{1}(M)) → M_{k}(Γ_{1}(N))  β_{d}(x) = ∑_{g∈R} gx  (1.6)

where the sum runs over a set of representatives R of Γ_{1}(N)\ ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix}) Γ_{1}(M).

These maps mirror exactly the corresponding maps on the side of modular forms. On the side of modular forms, α_{d} : M_{k}(Γ_{1}(M)) → M_{k}(Γ_{1}(N)) a and β_{d} : M_{k}(Γ_{1}(N)) → M_{k}(Γ_{1}(M)). [Define more precisely.] We have the following compatibilities [which I haven’t checked]:

⟨α_{d}(f), x⟩ = ⟨f, α_{d}(x)⟩,
⟨f, β_{d}(x)⟩ = ⟨β_{d}(f), x⟩.

1.1.4 Manin symbols

Consider the set of right cosets Γ\ SL_{2}(Z). The set of Manin symbols is

{0, …, k − 2} × (Γ\ SL_{2}(Z)).

Denote the Manin symbol corresponding to an integer i and a coset Γr by [i, Γr]. Define the action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Q) on the free abelian group on Manin symbols by

g.[i, Γr] = [g^{−1}(X^{i}Y^{k−2−i}), Γrg] = [(aX + bY)^{i}(cX + dY)^{k−2−i}, Γrg].

The right hand side is written as a sum of Manin symbols by expanding out (aX +bY)^{i}(cX +dY)^{k−2−i} and using the relation [P +Q, Γr] = [P, Γr]+[Q, Γr].

Let σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ). Let M_{k}(Γ)′ be the torsion free quotient of the abelian group generated by the finitely many symbols

{x = [i, Γr] : 0 ≤ i ≤ k − 2, r ∈ SL_{2}(Z)}

modulo the relations

x + σ.x = 0,  (1.7)
x + τ.x + τ^{2}.x = 0.  (1.8)

Theorem 1.4. We have an isomorphism M_{k}(Γ)′ ⟶ M_{k}(Γ) given by

[i, Γr] ↦ r(X^{i}Y^{k−2−i}{0, ∞}).

Conversion from modular to Manin symbols

By (1.4) we can go between Manin and modular symbols. Given a Manin symbols [i, Γr] with r = ( \begin{matrix}a & b \\ c & d \end{matrix} ), we have

[i, Γr] ⟷ r(X^{i}Y^{k−2−i}\{0, ∞\}) = (dX − bY)^{i}(−cX + aY)^{2−k−i} \{\frac{b}{d}, \frac{a}{c}\} .

Suppose, on the other hand, that we are given a modular symbol P (X, Y){α, β} and wish to find a sum of Manin symbols mapping to it. This is important from an algorithm point of view because it is sometimes more efficient to compute the action of an operator on the modular symbol representation than on the Manin symbols representation. A case in point is the Atkin-Lehner involutions w_{q} and new-old maps α_{d} and β_{d} which are all quite easy to compute on the level of modular symbols.

1.2 Method of graphs

1.3 Arithmetic of quaternion algebras

Modular algorithms

2.1 Basis of eigenforms 2.2 Modular degree 2.3 Congruences 2.4 Period lattices 2.5 Rational parts of special values of L-functions 2.6 Component groups 2.7 Implementation

Motivating problems

3.1 Visisibility of Shafarevich-Tate groups and

the Birch and Swinnerton-Dyer conjecture 3.2 Modularity of a complex A_{5} Galois represen-

tation 3.3 Rational points on X_{0}(p) over Q(μ_{p}) 3.4 Serre’s conjecture modulo pq?

Chapter 4 Tables

11

Bibliography

[BK75] B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476.

[Cre97] J.E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[Hij74] H. Hijikata, Explicit formula of the traces of Hecke operators for Γ_{0}(N), 56–82.

[Koh98] D. Kohel, Hecke module structure of quaternions.

[Mer94] L. Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations (Berlin), Springer, 1994, pp. 59–94.

[Mes86] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata) (1986), 217–242.

[Piz76a] Arnold Pizer, On the arithmetic of quaternion algebras, Acta Arith. 31 (1976), no. 1, 61–89.

[Piz76b] Arnold Pizer, On the arithmetic of quaternion algebras. II, J. Math. Soc. Japan 28 (1976), no. 4, 676–688.

[Piz76c] Arnold Pizer, The representability of modular forms by theta series, J. Math. Soc. Japan 28 (1976), no. 4, 689–698.

[Piz80] Arnold Pizer, An algorithm for computing modular forms on Γ_{0}(N), J. Algebra 64 (1980), no. 2, 340–390.