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Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 December 2009

Dmitry Faifman  and
Zeév Rudnick
Dmitry Faifman
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: [email protected])
Zeév Rudnick
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: [email protected])
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Abstract

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We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

Keywords

hyperelliptic curverandom matrix theorycentral limit theoremzeros of L-functions

MSC classification

Secondary: 11G20: Curves over finite and local fields 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 81 - 101
Copyright
Copyright © Foundation Compositio Mathematica 2009

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