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The Sato-Tate conjecture for a Picard curve with complex multiplication (with an appendix by Francesc FitĂ©). (English) Zbl 1440.11100
Lario, Joan-Carles (ed.) et al., Number theory related to modular curves: Momose memorial volume. Proceedings of the Barcelona-Boston-Tokyo Number Theory Seminar in memory of Fumiyuki Momose, Barcelona, Spain, May 21–23, 2012. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 701, 151-165 (2018).
Summary: Let \(C/\mathbb {Q}\) be the genus \(3\) Picard curve given by the affine model \(y^3=x^4-x\). In this paper we compute its Sato-Tate group, show the generalized Sato-Tate conjecture for \(C\), and compute the statistical moments for the limiting distribution of the normalized local factors of \(C\).
For the entire collection see [Zbl 1384.11002].

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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