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

By Dmitry Faifman, Z. Rudnick. Compositio Mathematica, 2008

48 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65cd39dcd71513f4b7625ce32a09fb1bad31dfcc

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Traces of high powers of the Frobenius class in the hyperelliptic ensemble (2008, background): “Our goal is to explore the opposite limit, that of fixed constant field and large genus (q fixed, g → ∞), cf [13, 6].”
The mean value of L(12,χ) in the hyperelliptic ensemble (2012, methodology): “The calculations presented here to establish an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field are based on elementary estimates and techniques which are in the spirit of those used by Faifman and Rudnick [6], Kurlberg and Rudnick [15] and”
Mod-Gaussian convergence: new limit theorems in probability and number theory (2008, background): “However, Faifman and Rudnick [5] and Kurlberg and Rudnick [13] have recently given examples of problems where it is possible to understand the limit g → +∞ for fixed q.”
Conjectures for the integral moments and ratios of L-functions over function fields (2014, result): “[FR] D. Faifman and Z. Rudnick, Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field, Compositio Mathematica 146(2010), 81–101.”
The fluctuations in the number of points on a hyperelliptic curve over a finite field (2008, methodology): “4. Finally, we refer to the recent preprint of Faifman and Rudnick [ 4 ] which studies the statistics of the zeros of the zeta function of the curves C F over a fixed finite field in the limit of large genus.”