Homework Assignment 9

(Math 252: Modular Abelian Varieties)

William A. Stein

Nov. 12 (Due: Nov. 19))

In this homework assignment, you will design and create a T-shirt for me.

Definition 1.1 (Congruence Graph). Let N be a positive integer. Define a graph as follows:

• Vertices: Gal(\overline{Q}/Q)-conjugacy classes of newforms in S_{2}(Γ_{0}(N)) of level dividing N. For example, if N = 33, there are exactly two vertices, corresponding to the newform of level 11 and the newform of level 33. If N = 43, there are also two vertices, corresponding to the degree 2 newform and the newform with rational Fourier coefficients.

• Edges: Say that two newforms f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} are “congruent modulo a prime ideal ℘” of the ring generated by the coefficients of f and g if a_{n} − b_{n} ∈ ℘ for all n ≤ [PSL_{2}(Z) : Γ_{0}(N)/ ± 1]/6 with (n, N) = 1. If f and g are nonconjugate newforms, which correspond to vertices of the graph, put an edge between them and label it p whenever f and g are congruent modulo a prime ℘ of residue characteristic p.

Definition 1.2 (Intersection Graph). Let N be a positive integer. To each newform f of level N_{f} dividing N, there is a corresponding abelian subvariety A_{f} ⊂ J_{0}(N) of dimension equal to the degree of f times the number of divisors of N/N_{f}. Define a graph as follows:

• Vertices: The abelian varieties A_{f}.

• Edges: Draw an edge between A_{f} and A_{g} whenever A_{f} ∩ A_{g} ≠ 0, and label it with the structure of the group A_{f} ∩ A_{g}.

The T-Shirt. Get together with the other students who turn in homework, and have each person choose a different number from the following list: 88, 125, 225, 389, or 431. Call this number M. Create a T-shirt for me:

• Back: The graph from Definition 1.1 associated to the collection of newforms of level dividing M.

• Front: The graph from Definition 1.2 associated to the collection of abelian variety factors of J_{0}(M) that correspond to newforms of level dividing M.

You should determine the combinatorial structure of your two graphs and lay them out on paper before drawing on the t-shirt. All the algorithms needed to do the computations for the homework assignment should be available in MAGMA – see

the IntersectionGroup and CongruenceGroupAnemic commands. If you can think

of a better way to present the combinatorial congruence or intersection data defined above, feel free to use it!