Effective equidistribution of eigenvalues of Hecke operators

Abstract

In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S(N, k) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator T p acting on S(N, k) of degree less than or equal to d. This allows us to determine an effectively computable constant B d such that if J 0 (N) is isogenous to a product of Q-simple abelian varieties of dimensions less than or equal to d, then N B d . We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general "all-purpose" equidistribution theorem.

Weighted Sato–Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)

“…As in [17], [21], and [25], we also obtain an effective version of Theorem 1.2 for N = 3, which gives the rate of convergence, but only for monomial functions. Its proof is based on the error term in the orthogonality relation (Equation 2).…”

Bounded gaps between primes in multidimensional Hecke equidistribution problems

“…This particular version is due to Murty and Sindha [21], though the version we use here is slightly weaker. Lemma 4.1 (Theorem 8 of [21]). If a sequence (a n ) of real numbers is equidistributed with respect to a probability measure µ in the interval [a, b] and e(t) = e 2πit , then for any x ≥ 2 and T ≥ 2,…”

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

“…Serre showed in [26] that d 2 (N ) is unbounded. His argument, based on the equidistribution of the eigenvalues of T acting on S 2 (N, C) for fixed prime and increasing N , was made effective by Murty-Sinha [22] and Royer [23], establishing an asymptotic bound of d 2 (N ) √ log log N as N → ∞. Theorem 1.4 allows us to improve this bound for certain values of N : Corollary 1.6.…”