Traces of high powers of the Frobenius class in the hyperelliptic ensemble

By Z. Rudnick. 2008

70 citations recorded by the source corpus.

The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/33804f2ee13168b3653815809b1c658670d97cec

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Citations in the corpus

Conjectures for the integral moments and ratios of L-functions over function fields (2014, background): “[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 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”