Local statistics for zeros of Artin--Schreier $L$-functions

Speaker: Alexei Entin

Date: Fri, Jul 29, 2022

Location: PIMS, University of Northern British Columbia

Conference: Moments of L-functions Workshop

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory


We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).
We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.
We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.

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