Explicitly Computing With Modular Abelian Varieties

William Stein Harvard University February 6, 2004

Center for Communications Research in Princeton

Overview of Talk

1. Modular Abelian Varieties

2. Computing With Modular Abelian Varieties

Modular Abelian Varieties _{Abel}

Abelian variety : A complete group variety

Examples:

1. Elliptic curves, e.g., y^{2} = x^{3} + ax + b 2. Jacobians of curves

3. Quotients of Jacobians of curves

The Modular Curve X_{1}(N) _{Hecke}

Let 𝔥^{∗} = {z ∈ C : ℑ(z) > 0} ∪ P^{1}(Q).

1. X_{1}(N)_{C} = Γ_{1}(N)\𝔥^{∗} (compact Riemann surface)

2. X_{1}(N) has natural structure of algebraic curve over Q

3. X_{1}(N)(C) = {(E, P) : ord (P) = N}/ ∼ (moduli space)

\begin{matrix}N & ≤ 10 & 11 & 13 & 37 & 169 & 512 & 2003 \\ genus(X_{1}(N)) & 0 & 1 & 2 & 40 & 1070 & 7809 & 166167 \end{matrix}

Modular Forms _{Hecke}

1. Cuspidal modular forms (of weight 2):

S_{2}(N) = H^{0} (X_{1}(N), Ω_{X_{1}(N)}^{1} )

2. f ∈ S_{2}(N) has Fourer expansion in terms of q(z) = e^{2πiz}

f = ∞ ; ∑ ; n=1 a_{n}q^{n}

3. Hecke algebra (commutative ring):

T = Z[T_{1}, T_{2}, …] ↪ End (S_{2}(N))

The Modular Jacobian J_{1}(N) _{Jacobi}

1. Jacobian of X_{1}(N):

J_{1}(N) = Jac (X_{1}(N))

2. J_{1}(N) is an abelian variety over Q of dimension g(X_{1}(N)).

3. The elements of J_{1}(N) parameterize divisor classes on X_{1}(N) of degree 0.

Modular Abelian Varieties _{Shimura}

A modular abelian variety A over a number field K is any abelian variety A (over K) such that there is a homomorphism

A → J_{1}(N)

with finite kernel.

Examples and Conjectures

Suppose dim A = 1.

• Theorem (Wiles, Breuil, Conrad, Diamond, Taylor). If K = Q then A is modular.

• Theorem (Shimura). If A has CM then A is modular.

• Definition: A over \overline{Q} is a Q-curve if for each Galois conjugate A^{σ} of A there is an isogeny A → A^{σ} (an isogeny is a map with finite kernel). Conjecture (Ribet, Serre). Over \overline{Q} the non-CM modular elliptic curves are exactly the Q-curves.

GL_{2}-type _{Ken Ribet}

Defn. A simple abelian variety A/Q is of GL_{2}-type if

End_{0}(A/Q) = End (A/Q) ⊗ Q

is a number field of degree dim (A).

Shimura associated GL_{2}-type modular abelian varieties to Teigenforms:

f = q + _{n≥2}^{∑} a_{n}q^{n} ∈ S_{2}(N)
I_{f} = Ker (T → Q(a_{1}, a_{2}, a_{3}, …)), T_{n} ↦ a_{n}

Abelian variety A_{f} over Q of dim = [Q(a_{1}, a_{2}, …) : Q]:

A_{f} := J_{1}(N)/I_{f}J_{1}(N)

Theorem (Ribet). Shimura’s A_{f} is Q-isogeny simple since

End_{0}(A_{f}/Q) = Q(a_{2}, a_{3}, …).

Also there is an isogeny J_{1}(N) ∼ ∏_{f} A_{f}, where the product is over Galois-conjugacy classes of f.

Conjecture. (Ribet)

The simple modular abelian varieties A over Q are exactly the simple abelian varieties over Q of GL_{2}-type.

Ribet proved that his conjecture follows from Serre’s unproven conjectures on modularity of odd mod p Galois representations.

2. Computing With Abelian Varieties

Goal: Develop a systematic theory for computing with modular abelian varieties.

Basic Problems: Presentation, isogeny testing, isomorphism testing, endomorphism ring, enumeration.

Arithmetic Problems: Special values of L-functions, computing Shafarevich-Tate groups, Tamagawa numbers, enumerating elements of isogeny class.

Presentation

Modular abelian varieties could be specified in many ways:

- Equations

- Built from newform abelian varieties A_{f}

- Arise theoretically (e.g., Jacobians of Shimura curves).

For all our questions we view a modular abelian variety as being defined in the following way. Any modular abelian variety B can be obtained by quotienting an abelian subvariety A ⊂ J_{1}(N) by a finite subgroup G. Thus we represent B by giving A and G ⊂ A.

Specifing A and G

An inclusion φ : A ↪ J_{1}(N) induces an inclusion on homology

H_{1}(A, Q) ↪ H_{1}(J_{1}(N), Q),

and A is completely determined by the image of H_{1}(A, Q) in the vector space H_{1}(J_{1}(N), Q).

We give A by giving a subspace V ⊂ H_{1}(J_{1}(N), Q). By the Abel-Jacobi theory there is a canonical isomorphism

J_{1}(N)(C) ≅ H_{1}(J_{1}(N), R)/ H_{1}(J_{1}(N), Z).

Likewise A(C) ≅ V_{R}/V_{Z}, where V_{Z} = V ⋂ H_{1}(J_{1}(N), Z), so

A(C)_{tor} ≅ V_{Q}/V_{Z}.

We give G by giving finitely many elements of V_{Q}/V_{Z}.

Recognition Problem

Problem: When does a subspace V ⊂ H_{1}(J_{1}(N), Q) correspond to an abelian subvariety A of J_{1}(N) over K?

Solution: Given an isogeny decomposition of J_{1}(N) over K as a direct sum of simple abelian varieties, I have an algorithm to solve this problem. Also, it is now straightforward to compute such a decomposition when K = Q. (Related: Find intersection of all abelian variety subspaces that contain V.)

Problem: Given a group G defined by a finite list of elements of V_{Q}/V_{Z}, find the smallest number field over which G is defined. This is important because if G is defined over K, then B = A/G is defined over K.

Solution??: I have not solved this problem.

Modular Symbols _{Manin}

Modular symbols provide a presentation of

H_{1}(X_{1}(N), Z)

on which one can give formulas for Hecke and other operators. They have been intensively studied by Birch, Manin, Shokurov, Mazur, Merel, Cremona, and others.

> M := CuspidalSubspace(ModularSymbols(Gamma1(11)));
> Basis(M);
[
 -1/5*{-1/2, 0} + -2/5*{-1/4, 0} + 3/5*{-1/7, 0} + -1/5*{7/15,1/2},
 -2/5*{-1/2, 0} + 1/5*{-1/4, 0} + 1/5*{-1/7, 0} + -2/5*{7/15,1/2}
]

Enumeration Problem Over Q

Problem: Give an algorithm to systematically enumerate every modular abelian variety over Q.

The isogeny classes of simple modular abelian varieties over Q are in bijection with newforms, which are eigenvectors for Hecke operators in the space S_{2}(Γ_{1}(N)) of modular forms. Using the Atkin-Lehner-Li theory of newforms, modular symbols, and linear algebra, we can thus enumerate the isogeny classes over Q.

I do not know how to find all elements of an isogeny class, except when A has dimension 1, where it is solved. Maybe find several by intersecting A ⊂ J_{1}(N) with other abelian varieties over Q, quotienting out by intersection, and proving quotient is not isomorphic to A.

Example

> Factorization(J1(17));
[*
<Modular abelian variety 17A of dimension 1, level 17
 and conductor 17 over Q, [
    Homomorphism from 17A to J1(17) given on integral
    homology by:
    [-3  1  2 -2  0 -2  2 -1  2  4]
    [-2 -2  0  0  0  0  0  2  4  0]
]>,
<Modular abelian variety 17A[2] of dimension 4, level 17
 and conductor 17^4 over Q, [
    Homomorphism from 17A[2] to J1(17) (not printing
    8x10 matrix)
]>
*]

Enumeration Problem Over \overline{Q}

Problem: Give an algorithm to systematically enumerate every modular abelian variety over \overline{Q}.

There is a huge amount of work by Shimura, Ribet, González, Lario, and others, but still nobody has given an algorithm to enumerate all isogeny classes of modular abelian varieties over \overline{Q} explicitly. By explicit, I mean in the sense of giving defining data, i.e., a pair (V, G ⊂ V_{Q}/V_{Z}).

Obstructions:

- Difficulty of constructing End (A_{f}/\overline{Q}) explicitly (I have an algorithm, but it is way too slow to be useful)

- Difficulty of decomposing A_{f}/\overline{Q} as a product of simples, even given End (A_{f}/\overline{Q}). Need a good “Meataxe” over Q.

Computing Endomorphism Rings

Problem: Given a modular abelian variety A over K, compute End (A) explicitly, i.e, give matrices in End (V) that generate End (A) as an abelian group.

Solution: When A ⊂ J_{1}(N) is simple, End (A) ⊗ Q is a skew field, which can be computed. For example, if K = Q, then A = A_{f} is attached to a newform and End (A) ⊗ Q is generated by the image of the Hecke algebra. We can then find End (A) in End (A) ⊗ Q as the Z-submodule of elements that preserve the lattice V_{Z}.

We can also explicitly compute Hom (A, B) for any modular abelian varieties A and B, by writing A and B as simples, computing endomorphism algebras, and finding the Z-module of homomorphisms that induce a map that fixes integral homology.

Example

> A := J0(33); A;
Modular abelian variety J0(33) of dimension 3 and level 3*11 over Q
> End(A);
Group of homomorphisms from J0(33) to J0(33)
> Basis(End(A));
[
    Homomorphism from J0(33) to J0(33) (not printing 6x6 matrix),
    Homomorphism from J0(33) to J0(33) (not printing 6x6 matrix),
    Homomorphism from J0(33) to J0(33) (not printing 6x6 matrix),
    Homomorphism from J0(33) to J0(33) (not printing 6x6 matrix),
    Homomorphism from J0(33) to J0(33) (not printing 6x6 matrix)
]
> Matrix(Basis(End(A))[2]);
[ 0  1  0  0  0 -1]
[ 0  1  0  0  0  0]
[ 0  1  0  0 -1  0]
[ 0  1 -1  1 -1  0]
[ 0  1 -1  0  0  0]
[-1  1  0  0  0  0]

Isogeny Testing

Problem: Given modular abelian varieties A and B, determine whether or not A is isogenous to B.

Determine whether A is isogenous to B is easy, since we may assume A and B are attached to newforms ∑ a_{n}q^{n} and ∑ b_{n}q^{n}, and then A is isogenous to B if and only if the newforms are Galois conjugate.

Isomorphism Testing

Problem: Suppose A is isogenous to B. Decide whether A is isomorphic to B.

I think this is an open problem. Assume we have computed End (A), End (B), and Hom (A, B) explicitly (we can). Given a basis for Hom (A, B), how do you know if some linear combination of that basis has determinant 1? It’s not clear (to me).

If A and B are both simple and have commutative endomorphism ring, then I found an algorithm to decide whether A is isomorphic to B. This algorithm can be extended to abelian varieties that are products of such A, assuming the factors occur with multiplicity one (up to isogeny). However, I do not know how to decide whether A^{n} is isomorphic to B^{n}, though I have a vague strategy that I think might work.

Algorithm for Testing Isomorphism

Suppose A and B are explicitly defined modular abelian varieties over Q that are both isogenous to an abelian variety A_{f}. The following algorithm determine whether A is isomorphic to B.

Let H = Hom (A, B). Both A and B are given explicitly by pairs (V, G_{1}) and (V, G_{2}), so we can compute an isogeny f : B → A. Let H_{f} = {ϕ ∘ f : ϕ ∈ H} ⊂ End (B). Note that A is isomorphic to B if and only if H_{f} contains an element of degree deg (f). Also note that H_{f} has finite index in End (B).

By hypothesis K = End (B) ⊗ Q is the field generated by the Fourier coefficients of f. The norm of an element of K is the positive square root of the degree of the corresponding homomorphism (see Milne in Cornell-Silverman, pg 126, Prop. 12.12).

Thus if deg (f) is not a perfect square, then there can be no element of B of degree deg (f), so A is not isomorphic to B. Thus suppose deg (f) = d^{2}.

Typically there will be infinitely many element in O_{K} of norm d, but there are only finitely many up to units. There is an algorithm, which involves computing the class group of O_{K}, which enumerates representive elements of O_{K} of norm d, up to units (e.g., the NormEquation command in MAGMA). Thus suppose we have computed representative elements z_{1}, …, z_{n} of the elements of O_{K} with norm d. Then A is isomorphic to B if and only if there is a unit u and a z_{i} such that u^{−1}z_{i} ∈ H_{f} ⊂ K. Equivalently, such that z_{i} ∈ uH_{f}. There are only finitely many possibilities for uH_{f}, since H_{f} has finite index in O_{K} and [O_{K}, uH_{f}] = [O_{K} : H_{f}], since O_{K} = uO_{K}. We can thus list all subgroups uH_{f} (since we can compute generaturs for O_{K}^{∗}) and hence determine whether H_{f} contains an element of norm d, as required.

Thank you for

inviting me!

Acknowledgements:

Papers of Ken Ribet, Goro Shimura, and Joan Lario. Conversations with John Cannon, Enrique Gonzalez, David Kohel, Barry Mazur, Allan Steel