Homework Assignment 2 Math 252: Modular Abelian Varieties

William A. Stein

Sep. 24 (Due: Oct. 1)

1. Prove that 𝔥^{∗} = 𝔥 ∪ Q ∪ {∞} is Hausdorff.

2. Compute the length of the geodesic path from −1 to 1 in the upper half plane with respect to the Poincaré metric. (It’s OK to compute a numerical approximation—I just want you to play around.)

3. (a) Suppose E = C/Λ is an elliptic curve and λ : C → C defines an automorphism of E. Prove that λ lies in a quadratic imaginary extension of Q.

(b) Suppose E is an elliptic curve and Aut(E) ≠ {±1}. Prove that E is isomorphic to E_{τ} with τ = i or τ = e^{2πi/3}.

4. Let Γ be a congruence subgroup of SL_{2}(Z), and let X = Γ\𝔥^{∗} be the corresponding compact Riemann surface. Prove that the degree of the natural map X → X(1) equals the index in PSL_{2}(Z) of the image of Γ in PSL_{2}(Z).

5. Find explicit basis for each of the following homology groups, along with all explicit natural maps you can think of between these groups:

H_{1}(X_{0}(11), Z), H_{1}(X_{0}(11), cusps, Z),
H_{1}(X_{1}(11), Z), H_{1}(X_{1}(11), cusps, Z).

6. If you read Lemma 1.41–1.42 on page 23 of Shimura’s book you’ll find a way to compare the cusps for Γ(N), i.e., the orbits for the action of Γ(N) on P^{1}(Q). Shimura proves that two cusps a/b and c/d (reduced fractions) are equivalent if and only if ±(a, b) = (c, d) (mod N).

(a) Use Shimura’s result to prove that the cusps for Γ(N) are in bijection with the following set: The vectors ±(a, b), where a, b ∈ Z/NZ and gcd(a, b, N) = 1, and the ± means that we identify (a, b) with (−a, −b).

(b) Deduce that for N > 2, the number of Γ(N) cusps is N^{2}/2·∏_{p|N}(1−p^{−2}).