  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.
