Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups

2025/07/03 by Gallese, Andrea, Goodson, Heidi, Lombardo, Davide · 2 citations
#11F80 #11G10 #11G15 #14C25 #14K15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.02535

Abstract

Denote by J_m the Jacobian variety of the hyperelliptic curve defined by the affine equation y²=x^m+1 over ℚ, where m ≥ 3 is a fixed positive integer. In this paper, we compute the Sato-Tate group of J_m. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and p-adic gamma functions at rational arguments.

Citations

• Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
• Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties
• Sato-Tate Distributions of Catalan Curves
• Sato-Tate groups of abelian threefolds
• Sato-Tate Distributions of y²=x^p-1 and y²=x^2p-1
• The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture of Banaszak and Kedlaya
• The Sato-Tate conjecture for a Picard curve with Complex Multiplication
• On the \ℓ-adic Galois representations attached to nonsimple abelian\n varieties
• On the Sato-Tate conjecture for non-generic abelian surfaces
• An algebraic Sato-Tate group and Sato-Tate conjecture

Cited by

• Degeneracy and Sato-Tate groups of y²=x^p2-1
• Connected monodromy fields of Jacobians with complex multiplication

Related

• Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
• Sato-Tate groups of genus 2 curves
• Degeneracy and Sato-Tate groups of y²=x^p2-1
• An algebraic Sato-Tate group and Sato-Tate conjecture
• The twisting Sato-Tate group of the curve y² = x⁸ - 14x⁴ + 1