Local statistics for zeros of Artin-Schreier 𝐿-functions

Entin, Alexei; Pirani, Noam

doi:10.1090/tran/8850

Local statistics for zeros of Artin-Schreier $L$-functions
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by Alexei Entin and Noam Pirani;

Trans. Amer. Math. Soc. 376 (2023), 6141-6175

DOI: https://doi.org/10.1090/tran/8850

Published electronically: June 29, 2023

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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.

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