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Sato-Tate distributions. (English) Zbl 1440.11176

Bucur, Alina (ed.) et al., Analytic methods in arithmetic geometry. Arizona winter school 2016, the University of Arizona, Tucson, AZ, USA, March 12–16, 2016. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (CRM). Contemp. Math. 740, 197-248 (2019).
Summary: In this expository article we explore the relationship between Galois representations, motivic \(L\)-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension \(g\leq 3\) and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry.
For the entire collection see [Zbl 1435.11006].

MSC:

11M50 Relations with random matrices
11G10 Abelian varieties of dimension \(> 1\)
11G20 Curves over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K15 Arithmetic ground fields for abelian varieties

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