Homework Assignment 3 Math 252: Modular Abelian Varieties

William A. Stein

Oct. 1 (Due: Oct. 8)

There are four problems, each of equal weight.

1. Let p ≥ 5 be a prime. In this problem, we apply a general result of Shimura to show that the genus of X_{1}(p) is (p − 5)(p − 7)/24. Let \overline{Γ}_{1}(N) denote the image of Γ_{1}(N) in PSL_{2}(Z).

(a) Read Prop. 1.40 of Shimura’s Introduction to the Arithmetic Theory of Automorphic Functions, which asserts that for a finite index subgroup G of PSL_{2}(Z), the genus of X(G) is

g = 1 + \frac{μ}{12} − \frac{ν_{2}}{4} − \frac{ν_{3}}{3} − \frac{ν_{∞}}{2} ,

where μ = [PSL_{2}(Z) : G], ν_{2} is the number of elliptic points for G of order 2, ν_{3} is the number of elliptic points for G of order 3, and ν_{∞} is the number of cusps. An elliptic point of order n > 1 is an element z ∈ 𝔥 whose stabilizer in G has order n. [Hint: You do not have to actually write anything at all to get full credit for this part of the problem!]

(b) Let G = \overline{Γ}_{1}(N). Show that μ = (p^{2} − 1)/2. [Hint: Use a past homework problem.]

(c) Let G = \overline{Γ}_{1}(N). Show that ν_{∞} = p − 1. [Hint: From a past homework problem, you know that the cusps for Γ(p) are in bijection with vectors ±(x, y), where x, y, ∈ Z/pZ and gcd(x, y, p) = 1. The cusps for Γ_{1}(p) are the orbits of the cusps for Γ(p) under the action of ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Show that every such orbit contains an element (x, y) with 0 ≤ y < p and x = 0.]

(d) Let G = \overline{Γ}_{1}(N), with N ≥ 5. Show that ν_{2} = ν_{3} = 0. [Hint: Shimura proves the analogue of this statement in his book. You could find his proof and adapt it.]

(e) Conclude that if p ≥ 5 is prime, then the genus of X_{1}(p) is (p−5)(p−7)/24.

2. For i = 1, 2, let g_{1} = ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ) ∈ \overline{Γ}_{1}(N). Then g_{1} and g_{2} lie in the same right

coset \overline{Γ}_{1}(N)x if and only if c_{1} ≡ εc_{2} (mod N) and d_{1} ≡ εd_{2} (mod N), where ε = ±1. [Hint: Compute g_{1}g_{2}^{−1} explicitly to obtain a system of congruences that characterize whether or not g_{1} and g_{2} are in the same coset.] 3. For i = 1, 2 let α_{i} = a_{i}/b_{i} be cusps written in lowest terms. Then α_{2} =

γ(α_{1}) for some γ ∈ \overline{Γ}_{1}(N) if and only if b_{2} ≡ εb_{1} (mod N) and a_{2} ≡ εa_{1} (mod gcd(b_{1}, N)), with ε = ±1. [Hint: The tricky part is to prove that the congruence conditions imply that the cusps are conjugate; for this, construct matrices ( \begin{matrix}a_{2} & r_{2} \\ b_{2} & s_{2} \end{matrix} ) and ( \begin{matrix}a_{1} & r_{1} \\ b_{1} & s_{1} \end{matrix} ) that, by Problem 2, lie in the same right coset.]

4. In this problem we compute with modular symbols for the modular curve X_{1}(5).

Recall that

s = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) and t = (\begin{matrix}1 & −1 \\ 1 & 0 \end{matrix}) .

(a) Show that the genus of X_{1}(5) is 0 using Problem 1.

(b) Use Exercise 2 to write down (p^{2} − 1)/2 = 12 pairs (c, d) that are the bottom two entries of a set of right coset representatives for \overline{Γ}_{1}(5) in PSL_{2}(Z).

(c) Let C be the free abelian group generated by the pairs (c, d) from part (ii), subject to the relations x + xs = 0 and x = 0 if x = xs, where x runs through the 12 pairs (c, d). What is the rank of C?

(d) Compute the kernel Z of the linear map C → Div(X_{1}(5)) that sends (c, d) to g(∞) − g(0) ∈ Div(X_{1}(5)), where g = ( \begin{matrix}∗ & ∗ \\ c & d \end{matrix}).

(e) Compute the subgroup B of C generated by x + xt + xt^{2} and also x when x = xt, where x runs through the 12 pairs (c, d).

(f) Observe that B = Z, so H_{1}(X_{1}(5), Z) = 0.

(g) Compute C/B. Remark: (not part of the problem) The group C/B is isomorphic to the relative homology

H_{1}(X_{1}(5), Z, {cusps}).

(h) Challenge (“extra credit”): Prove that for any prime p, if we construct C and Z as above, but with Γ_{1}(5) replaced by Γ_{1}(p), then the quotient C/Z is free of rank p − 2.