Hyperelliptic curves, L-polynomials, and random matrices

Details Hyperelliptic curves, L-polynomials, and random matrices Hyperelliptic curves, L-polynomials, and random matrices

2008-03-31

arxiv.org/abs/0803.4462v4

Arithmetic, Geometry, Cryptography and Coding Theory (AGCT-11, 2007), Contemporary Mathematics volume 487, pp. 119-162, AMS, 2

Mathematics

Number Theory

Fixed 3 minor typos on pages 31 and 32, including a correction to Table 12. 44 pages

Scientific paper

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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