Connected monodromy fields of Jacobians with complex multiplication

2025/10/24 by Gallese, Andrea, Lombardo, Davide
#11Y40 #14C30 #14K20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11F80 #Secondary: 11G15

paper · doi:10.48550/arxiv.2510.21247

Abstract

We describe an algorithm to compute the minimal field of definition of the Tate classes on powers of a Jacobian J with potential complex multiplication. This field arises as a natural invariant of the Galois representations attached to J. We also give closed formulas expressing the periods of anti-holomorphic differential forms on J in terms of the periods of the holomorphic ones.

Citations

• Mumford-type Shimura curves contained in the Torelli locus
• Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups
• Monodromy groups of Jacobians with definite quaternionic multiplication
• Non-isogenous abelian varieties sharing the same division fields
• Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms
• Computing period matrices and the Abel-Jacobi map of superelliptic\n curves
• An algebraic Sato-Tate group and Sato-Tate conjecture
• Jacobians with complex multiplication
• Galois representations
• Connectedness extensions for abelian varieties

Related

• Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
• Monodromy groups of Jacobians with definite quaternionic multiplication
• Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication
• Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties
• Jacobians among Abelian threefolds: a formula of Klein and a question of Serre