Average Distributions and Product of Special Values of L-Series

Let E be an elliptic curve defined over the rationals. For any prime p of good reduction, let Ep be the elliptic curve over Fp obtained by reducing E mod p. Let ap(E) be the trace of the Frobenius morphism of Ep. Then, Hasse proved that #E(Fp) = p + 1 − ap(E) with |ap(E)| ≤ 2 √ p. The case ap(E) = 0 corresponds to supersingular reduction mod p. Let N be a positive integer. For a fixed r ∈ Z, and fixed curves E1, . . . , EN , we define π E1,...,EN (x) = # {p ≤ x : ap(E1) = . . . = ap(EN) = r} . There is a simple heuristic that can be used to predict the asymptotic behavior of π E1,...,EN (x). From Hasse’s bound, the probability that ap(E) = r is