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Character theory approach to Sato–Tate groups

Part of: Abelian varieties and schemes Zeta and $L$-functions: analytic theory Arithmetic problems. Diophantine geometry Representation theory of groups Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 August 2016

Yih-Dar Shieh
Yih-Dar Shieh*
Affiliation:
Institut de Mathématiques de Marseille, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France email chiapas@gmail.com http://yih-dar.shieh.perso.luminy.univ-amu.fr/

Abstract

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In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points provide satisfactory convergence. Even for $g=2$ , the classical approach using moment statistics requires about $2^{30}$ sample points to obtain such information.

MSC classification

Primary: 11M50: Relations with random matrices
Secondary: 20C15: Ordinary representations and characters 11G10: Abelian varieties of dimension $> 1$ 11G20: Curves over finite and local fields 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 14K15: Arithmetic ground fields

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