Connected monodromy fields of Jacobians with complex multiplication

TL;DR

This paper presents an algorithm to determine the minimal field of definition for Tate classes on Jacobians with potential complex multiplication and relates periods of differential forms to those of holomorphic forms.

Contribution

It introduces a new algorithm for computing the minimal field of Tate classes and provides formulas linking periods of anti-holomorphic and holomorphic differential forms.

Findings

Algorithm for minimal field of Tate classes on Jacobians with CM

Closed formulas relating periods of differential forms

Enhanced understanding of Galois representations of Jacobians

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.