Local Statistics for Zeros of Artin-Schreier L-functions

Abstract

We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L-functions: the ordinary, polynomial (the p-rank 0 stratum) and odd-polynomial families. We compute 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 a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T]$

“…The statistics for families of higher order characters (r > 2) are less well known. However, there has been some recent progress in studying them [3,7,10,15]. The author has already considered simpler statistics for a similar, more geometric family in [21], extending the results of [3].…”

“…( 3 ) There are specific counterexamples to this assumption arising from L-functions attached to extensions with quadratic subfields. See, for example, the case of odd Artin-Schreier curves in [15].…”