The fluctuations in the number of points on a hyperelliptic curve over a finite field

By Pär Kurlberg, Z. Rudnick. 2008

62 citations recorded by the source corpus.

The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\sqrt{q}$ is distributed as the trace of a random $2g\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\sqrt{q}$ has a standard Gaussian distribution.

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

View Paper: https://www.semanticscholar.org/paper/eac902120aafbeaae89f2947bf99b4e1032bb351

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In Algebra and Number Theory

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Citations in the corpus

Statistics for Traces of Cyclic Trigonal Curves over Finite Fields (2009, methodology): “We transcribe here the relevant results from the work of Kurlberg and Rudnick [5].”
Traces of high powers of the Frobenius class in the hyperelliptic ensemble (2008, unknown): “Since χQ(P d) = χQ(P ) for d odd, the average is 〈Hn〉 = − 1 (q − 1)q2g+ n2 ∑ d|n 3≤d odd n d ∑ deg P= n d ∑ 2α+β=2g+1 σn/d(α) ∑ deg B=β ( B P ) = − 1 (q − 1)q2g+ n2 ∑ d|n 3≤d odd n d ∑ 2α+β=2g+1 σn/d(α)S(β; n d ) .”
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.”
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”
Conjectures for the integral moments and ratios of L-functions over function fields (2014, result): “[KR] P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field, J. of Number Theory, 129(2009), 580–587.”