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arXiv · 0803.4462

Hyperelliptic curves, L-polynomials, and random matrices

Kiran S. Kedlaya·Andrew V. Sutherland
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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.

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Kiran S. Kedlaya, Andrew V. Sutherland. 2010-11-28. Hyperelliptic curves, L-polynomials, and random matrices. https://doi.org/10.1090/conm%2F487

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