| Summary: | We analyze the distribution of unitarized L-polynomials Lp(T)
(as p varies) obtained from a hyperelliptic curve of genus g [less than or equal to] 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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