Homework Assignment 8

(Math 252: Modular Abelian Varieties)

William A. Stein

Nov. 3 (Due: Nov. 12))

1. The principal congruence subgroup Γ(N) of level N is the kernel of the reduction map SL_{2}(Z) → SL(2, Z/NZ). The subgroup Γ_{1}(N) consists of matrices of the form ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) modulo N. Let Γ ⊂ SL_{2}(Z) be a subgroup that contains Γ(N) for some N. Show that there exists g ∈ GL(2, Q) such that the conjugate g^{−1}Γg, which is a subgroup of GL(2, Q), contains Γ_{1}(N^{2}). This shows that many problems about modular forms can be reduced to questions about modular forms for Γ_{1}.

2. Let f ∈ S_{k}(Γ_{1}(N)) be a nonzero modular form that is an eigenform for all the Hecke operators T_{p} and for the diamond bracket operators ⟨d⟩. Let

ε : (Z/NZ)^{∗} → C^{∗}

be the character of f, so ⟨d⟩f = ε(d)f for all d ∈ (Z/NZ)^{∗}. Show that f satisfies the following equation:

for any ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N),

f(z) = ε(d)(cz + d)^{−k}f(\frac{az + b}{cz + d}).

3. Let M be a positive integer and let p be a prime.

(a) Show that there are two injective linear maps

S_{2}(Γ_{1}(M)) ↪ S_{2}(Γ_{1}(pM))

sending f(q) to f(q) and sending f(q) to f(q^{p}). (I proved this in class, and you can copy the proof from your notes.)

(b) Is it ever the case that the intersection of the images of these two maps is nonzero?

4. Let M be an integer such that S_{2}(Γ_{1}(M)) has positive dimension, and let p be a prime (thus M = 11 or M ≥ 13).

(a) Let f ∈ S_{2}(Γ_{1}(M)) be an eigenvector for T_{p} with eigenvalue λ. Show that T_{p} acting on S_{2}(Γ_{1}(Mp)) preserves the two-dimensional subspace generated by f and f(pz). (When p divides the level, we define T_{p}(∑ a_{n}q^{n}) = ∑ a_{np}q^{n}.)

(b) Show that if λ^{2} ≠ 4p then T_{p} is diagonalizable on this 2-dimensional space. What are the eigenvalues of T_{p} on this space? In fact, one never has λ^{2} = 4p; see [Coleman-Edixhoven, On the semisimplicity of the U_{p} operator on modular forms] for more details.

(c) Show that for any r > 2, the Hecke operator T_{p} on S_{2}(Γ_{1}(Mp^{r})) is not diagonalizable.

(d) Deduce that for r > 2 the Hecke algebra T associated to S_{2}(Γ_{1}(Mp^{r})) has nilpotent elements, so it is not a subring of a product of number fields.