Sato-Tate Distributions of y²=x^p-1 and y²=x^2p-1

2020/04/22 by Emory, Melissa, Goodson, Heidi · 2 citations
#11G10 #11G20 #11M50 (primary) #14G10 (secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.10583

Abstract

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y²=x^p-1 and y²=x^2p-1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.

Cited by

• Degeneracy and Sato-Tate groups of y²=x^p2-1
• Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

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