2008/04/04 by Pär Kurlberg, Zeév Rudnick, Kurlberg, P. +1 · 1 citation
Computer Science · Mathematics · #11G25 #11L40 #Advanced Algebra and Geometry #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0804.0808
openalex publication_date 2008/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/√(q) is distributed as the trace of a random 2g× 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 ± 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/√(q) has a standard Gaussian distribution.