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Connected monodromy fields of Jacobians with complex multiplication

Andrea Gallese, Davide Lombardo

TL;DR

This work develops an explicit algorithm to compute the connected monodromy field $k(\varepsilon_A)$ for abelian varieties with complex multiplication, focusing on CM Jacobians. By combining CM theory with a detailed analysis of endomorphisms, CM-type, Mumford–Tate groups, and the Galois action on de Rham and Hodge structures, the authors reduce the problem to finitely many algebraic period relations $P(\sigma,\omega)$ and exploit algebraicity results for CM periods to determine $k(\varepsilon_A)$ (and its relation to $k(\operatorname{End}A)$). A key advance is the explicit link between anti-holomorphic and holomorphic periods via algebraic formulas, enabling practical computation of periods and the monodromy field in CM cases. The paper also provides an algorithmic framework, with computational steps and concrete examples (including Fermat and Mumford-type Jacobians), illustrating how the connected monodromy field is detected from period data and Galois actions. This yields concrete, CM-friendly methods to access Galois invariants of Tate/Hodge structures attached to powers of CM Jacobians and connects deep conjectures (e.g., MT) to tangible, computable invariants.

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.

Connected monodromy fields of Jacobians with complex multiplication

TL;DR

This work develops an explicit algorithm to compute the connected monodromy field for abelian varieties with complex multiplication, focusing on CM Jacobians. By combining CM theory with a detailed analysis of endomorphisms, CM-type, Mumford–Tate groups, and the Galois action on de Rham and Hodge structures, the authors reduce the problem to finitely many algebraic period relations and exploit algebraicity results for CM periods to determine (and its relation to ). A key advance is the explicit link between anti-holomorphic and holomorphic periods via algebraic formulas, enabling practical computation of periods and the monodromy field in CM cases. The paper also provides an algorithmic framework, with computational steps and concrete examples (including Fermat and Mumford-type Jacobians), illustrating how the connected monodromy field is detected from period data and Galois actions. This yields concrete, CM-friendly methods to access Galois invariants of Tate/Hodge structures attached to powers of CM Jacobians and connects deep conjectures (e.g., MT) to tangible, computable invariants.

Abstract

We describe an algorithm to compute the minimal field of definition of the Tate classes on powers of a Jacobian with potential complex multiplication. This field arises as a natural invariant of the Galois representations attached to . We also give closed formulas expressing the periods of anti-holomorphic differential forms on in terms of the periods of the holomorphic ones.
Paper Structure (13 sections, 11 theorems, 71 equations)

This paper contains 13 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Computing the endomorphism ring
  3. Computing the CM-type
  4. The CM type
  5. Computing the Mumford-Tate group
  6. Computing the algebra of Hodge classes
  7. Computing the Galois action on de Rham homology
  8. Computing the Galois action on Hodge cycles
  9. Computing the Galois action on Tate classes
  10. Computing the periods
  11. Algebraic relations between periods and quasi-periods
  12. Computational considerations
  13. Examples

Key Result

Theorem 1.3

Let $A/k$ be an abelian variety with complex multiplication. Let $\mathcal{F}$ be a finite generating set of equations for $\mathop{\mathrm{MT}}\nolimits(A)$. The extension $k(\varepsilon_A)/k(\mathop{\mathrm{\operatorname{End}}}\nolimits A)$ is abelian and corresponds to the subgroup of $\tau \in \

Theorems & Definitions (47)

  • Definition 1.2
  • Theorem 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 3.2
  • Remark 3.4
  • ...and 37 more