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Distribution of zeta zeroes of Artin-Schreier covers. (English) Zbl 1348.11048

Summary: We study the distribution of the zeros of the zeta functions of the family of Artin-Schreier covers of the projective line over \(\mathbb{F}_q\) when \(q\) is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of \([-\pi,\pi)\) has a standard Gaussian distribution (when properly normalized).

MSC:

11G20 Curves over finite and local fields
11M50 Relations with random matrices
14G15 Finite ground fields in algebraic geometry
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