Weighted distribution of low-lying zeros of GL(2) L-functions

2018/06/15 by Andrew Knightly, Knightly, Andrew, Caroline Reno +1 · 3 citations
Mathematics · #11F11 #11M26 #11M41 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1806.05869

openalex publication_date 2018/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if the zeros of an automorphic L-function are weighted by the central value of the L-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the L-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted L-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Cited by

• Centered Moments of Weighted One-Level Densities of GL(2) L-Functions
• Low-lying zeros of Hilbert modular L-functions weighted by powers of central L-values
• Low-lying zeros of symmetric power L-functions weighted by symmetric square L-values

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