The Sato-Tate conjecture for a Picard curve with Complex Multiplication
Joan-C. Lario, Anna Somoza (with an appendix by Francesc Fit\'e)
TL;DR
This paper investigates the Sato-Tate conjecture for a specific genus 3 Picard curve with complex multiplication, confirming the conjecture and analyzing the distribution of its local factors.
Contribution
It computes the Sato-Tate group for the curve, proves the generalized Sato-Tate conjecture, and determines the statistical moments of the local factors' distribution.
Findings
Sato-Tate group explicitly computed
Generalized Sato-Tate conjecture proven for the curve
Statistical moments of local factors calculated
Abstract
Let C/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.
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Taxonomy
TopicsVietnamese History and Culture Studies · Historical Studies and Socio-cultural Analysis · Algebraic Geometry and Number Theory