Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions

Abstract

In this paper we apply the L-function Ratios Conjecture to compute the one-level density for a symplectic family of L-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function f reaches 1, we observe a transition in the main term, as well as in the lower order term. Assuming GRH, we then directly calculate main and lower order terms for test functions f such that supp( f) ⊂ (−1, 1), and observe that the result is in agreement with the prediction provided by the Ratios Conjecture. As a corollary, we deduce that, under GRH, at least 75% of these L-functions do not vanish at the central point.

Low-lying zeros in families of holomorphic cusp forms: the weight aspect

“…In particular, when (m, n) = 1 this estimate is nontrivial in the range mn ≤ k 4−ε , whereas [ILS,Corollary 2.2] is nontrivial up to mn ≤ k 10 3 −ε . The terms involving φ (j) (1) in (1.3) are responsible for a sharp transition at 1 in these orthogonal families and are analogous to those obtained in symplectic families in [FI,FPS2,FPS3,R,Wax]. Indeed, in the family of real Dirichlet characters considered in [FPS2], after applying the explicit formula and treating the resulting sums over primes by repeatedly using the Poisson summation formula, one obtains lower-order terms involving φ (j) (1).…”

“…This work was inspired by the function field case considered in [R], in which, using Poisson summation, the 1-level density is turned into an average of the trace of the Frobenius class in the hyperelliptic ensemble, from which a transition term is isolated using the explicit formula. Transition terms also surface in predictions coming from the L-function Ratios Conjecture [FPS3,Wax]; in this case one needs to compute averages of ratios of local factors at infinity. In the current situation, these terms come from a significantly different source, namely from a careful analysis of averages of Bessel functions and Kloosterman sums coming from the Petersson trace formula.…”