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Weighted Distribution of Low-lying Zeros of GL(2) $L$-functions

Part of: Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  08 January 2019

Andrew Knightly  and
Caroline Reno
Andrew Knightly
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA Email: knightly@math.umaine.educaroline.reno@maine.edu
Caroline Reno
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, USA Email: knightly@math.umaine.educaroline.reno@maine.edu
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Abstract

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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).

Keywords

low-lying zeroL-function

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions
Secondary: 11F11: Holomorphic modular forms of integral weight 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 1 , February 2019 , pp. 153 - 182
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was partially supported by grant #317659 from the Simons Foundation to author A. K. Section 3 is based in part on the University of Maine MA thesis of author C. R.

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