Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Abstract

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

Convergence to the Plancherel measure of Hecke eigenvalues

“…One can extend Theorem 1 to the case of any fixed even weight k and N square-free with (mn, N ) = 1 using the same proof. Using a more flexible method to remove the harmonic weights in the Petersson formula, Petrow [Pet18] derives a slightly weaker but uniform and more general form of Theorem 1 in which the weights and nebentypus characters are allowed to vary. It is worth noting that while his exponent is worse, the range of n over which the bound is non-trivial is as strong as in Theorem 1.…”

“…This allows us to use the Petersson trace formula, replacing class numbers with Kloosterman sums, which enjoy sharp bounds coming from the arithmetic geometry of curves (see [Wei48]). This technique applied to a similar problem is outlined in [Sar02] and was used earlier on other problems by Iwaniec in [Iwa84] and [ILS00], and more recently in [Pet18]. It allows one to quadruple the exponent of n in (1), which is crucial for some of our applications.…”

The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound

“…[ILS] [PY2] [BM]), and it is not clear that there is any canonical choice for general level. Let ξ δ (d) be the coefficients defined in [P2,Prop. 7.1].…”

An explicit formula for the second moment of Maass form symmetric square $L$-functions

“…When L is a power of 2 and 4|M we can evaluate P ∞,∞ and P ∞,0 explicitly in terms of λ f (2) since in this case we need only the values of ξ g (d) for g and d being a power of 2. These values are given on p. 2490 of [44] and for ν ≥ 1 are equal to…”