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The Sato-Tate conjecture for a Picard curve with complex multiplication (with an appendix by Francesc Fité). (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
##### Keywords:
Sato-Tate conjecture; Picard curve
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##### References:
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