Homework Assignment 5

(Math 252: Modular Abelian Varieties)

William A. Stein

Oct. 15 (Due: Oct. 22)

There are two problems and parts of these two problems are of equal weight.

1. Let T = V/T be a complex torus, and T_{1} = V_{1}/T_{1} and T_{2} = V_{2}/T_{2} be subtori of T such that V_{1} ∩ V_{2} = 0. In this problem, you will learn how to compute T_{1} ∩ T_{2}, which will help you to understand how I designed the T-shirt with all the wavy lines for the MSRI conference on modularity of elliptic curves.

(a) Prove that there is an exact sequence

0 → T_{1} ∩ T_{2} → T_{1} ⊕ T_{2} → T

where the map T_{1} ⊕ T_{2} → T is (a, b) ↦ a − b.

(b) Prove that there is an exact sequence

0 → T_{1} ∩ T_{2} → T_{1} ⊕ T_{2} → T′ → 0

where T′ = T_{1} + T_{2} ⊂ T.

(c) Prove that T′ = (V_{1} + V_{2})/L′, where L′ = (V_{1} + V_{2}) ∩ L.

(d) Conclude that

T_{1} ∩ T_{2} ≅ L′/(L_{1} + L_{2}),

where L′ is the saturation of L_{1} + L_{2} in L (i.e., L′ = (V_{1} + V_{2}) ∩ L).

2. It is a fact (which you do not have to prove) that there is an abelian variety called J_{0}(37) = V/L that contains the following two elliptic curves as abelian subvarieties:

E_{1} :  y^{2} + y = x^{3} − x
E_{2} :  y^{2} + y = x^{3} + x^{2} − 23x − 50

There is a basis for L with respect to which the lattices L_{1} and L_{2} that define E_{1} and E_{2} have the following basis:

L_{1} :  [1, −1, 1, 0], [1, −1, −1, 1]
L_{2} :  [0, 0, 0, 1], [1, 1, 1, 0]

(a) Use problem 1 to show that E_{1} ∩ E_{2} = Z/2Z × Z/2Z.

(b) Part (a) shows that if a_{p}(E_{i}) = p + 1 − #E_{i}(F_{p}), then for every prime p, we have a_{p}(E_{1}) ≡ a_{p}(E_{2}) (mod 2). You do not have to prove this fact to get full credit on this problem (though you are welcome to try, using, e.g., the nontrivial fact that a_{p}(E_{i}) is the eigenvalue of the Hecke operator T_{p} acting on either of the basis vectors for L_{i}). Verify that a_{p}(E_{1}) ≡ a_{p}(E_{2}) (mod 2) for p ≤ 7.

(c) (*) By any means necessary, determine the rational numbers L(E_{i}, 1)/Ω_{E_{i}} for these two elliptic curves. Note that L(E_{1}, 1)/Ω_{E_{1}} is not congruent modulo 2 to L(E_{2}, 1)/Ω_{E_{2}}. (Two rational numbers a and b are congruent modulo a prime p if ord_{p}(a − b) > 0.) This example illustrates that although the coefficients that define the L-series are congruent modulo 2, the rational parts of their special values need not be.