Hyperelliptic curves, L-polynomials, and random matrices

TL;DR

This paper investigates the distribution of L-polynomials from hyperelliptic curves of genus up to 3, confirming predictions from random matrix theory and proposing a genus 2 Sato-Tate type conjecture with exceptional cases.

Contribution

It provides experimental validation of the Katz-Sarnak model for hyperelliptic curves and introduces a new Sato-Tate conjecture for genus 2 curves with detailed subgroup classifications.

Findings

Distribution of L-polynomials matches random matrix predictions

Formulation of a genus 2 Sato-Tate conjecture with 22 exceptional cases

Identification of curves matching each proposed distribution

Abstract

We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sarnak random matrix model) between the distributions of Lp(T) and of characteristic polynomials of random matrices in the compact Lie group USp(2g). We then formulate an analogue of the Sato-Tate conjecture for curves of genus 2, in which the generic distribution is augmented by 22 exceptional distributions, each corresponding to a compact subgroup of USp(4). In every case, we exhibit a curve closely matching the proposed distribution, and can find no curves unaccounted for by our classification.