an:05592921
Zbl 1233.11074
Kedlaya, Kiran S.; Sutherland, Andrew V.
Hyperelliptic curves, \(L\)-polynomials, and random matrices
EN
Lachaud, Gilles (ed.) et al., Arithmetic, geometry, cryptography and coding theory. Proceedings of the 11th international conference, CIRM, Marseilles, France, November 5--9, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4716-9/pbk). Contemporary Mathematics 487, 119-162 (2009).
2009
a
11G40 11G30 11M50
Sato-Tate conjecture; trace of Frobenius; zeta-function; Haar measure; moment sequence
Summary: We analyze the distribution of unitarized \(L\)-polynomials \(L_p(T)\) (as \(p\) varies) obtained from a hyperelliptic curve of genus \(g\leq 3\) defined over \(\mathbb Q\). In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sarnak random matrix model) between the distributions of \(L_p(T)\) and of characteristic polynomials of random matrices in the compact Lie group \(\text{USp}(2g)\). 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 \(\text{USp}(4)\). In every case, we exhibit a curve closely matching the proposed distribution, and can find no curves unaccounted for by our classification.
For the entire collection see [Zbl 1166.11003].