Random Modular Symbols

Abstract

Let Γ denote the subgroup Γ_0^± (N) of GL_2(Z), N prime. Let V be the space of holomorphic modular forms for Γ. Let V_α ⊂ V denote the various Hecke eigenspaces, with the last V_α denoting the Eisenstein subspace. If M ∈ V is a modular symbol, define the type of M to be (t_1,...,t_k,t_E) where t_α = 1 if the projection of M to V_α is nonzero, and t_α = 0 otherwise.

For each N ≤ 100, we compute the types of the modular symbols in an increasing series of concentric boxes. We prove an obstruction for a given type to occur, related to the existence of “Eisenstein primes.” For any given type that survives this obstruction, we give computational evidence that the proportion of its occurrence in a box stabilizes as the boxes grow larger. We interpret the limit of this ratio (assuming it exists) as the box size goes to infinity as the probability that a random modular symbol will have this type.

Contrary to our original expectation, it does not appear to be the case that with probability 1 a random symbol will project nontrivially to each Vi. Whether the limit referred to in the previous paragraph actually exists, and why the limits have the various values that appear in our computations, are open questions.