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Hyperelliptic curves, \(L\)-polynomials, and random matrices. (English) Zbl 1233.11074
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).
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].

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11M50 Relations with random matrices
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