Hyperelliptic curves, L-polynomials, and random matrices
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.
Cite
@article{arxiv.0803.4462,
title = {Hyperelliptic curves, L-polynomials, and random matrices},
author = {Kiran S. Kedlaya and Andrew V. Sutherland},
journal= {arXiv preprint arXiv:0803.4462},
year = {2013}
}
Comments
Fixed 3 minor typos on pages 31 and 32, including a correction to Table 12. 44 pages