Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A framework for Tate modules of abelian varieties under isogeny

Sarah Frei, Katrina Honigs, John Voight

TL;DR

The paper develops a category-theoretic, linear-algebraic framework for Tate modules of abelian varieties under isogeny by packaging torsion into the adelic module $\widehat{T}A$ and its rationalization $\widehat{V}A$, with Galois action given by $\rho_A$. The central contribution is a functor $\Lambda$ from the isogeny category to sublattices of $\widehat{V}A_0$, yielding an equivalence of categories and a natural identification $\ker\varphi \cong \mathrm{coker}~\widehat{T}(\varphi)$ that translates isogenies into lattice inclusions; this provides a computable dictionary between isogenous abelian varieties and $\mathrm{Gal}_K$-stable lattices. The framework enables explicit change-of-basis computations for Galois representations and polarizations, illustrated by an extended example showing how to determine $\ell$-adic actions on $A$ and on $A^{\vee}$ via polarization data. Together, these results furnish a structured toolkit for analyzing Galois images of torsion and Tate modules in arithmetic geometry, with practical matrix-calculation methods for isogeny-related questions.

Abstract

We explain the linear algebraic framework provided by Tate modules of isogenous abelian varieties in a category-theoretic way.

A framework for Tate modules of abelian varieties under isogeny

TL;DR

The paper develops a category-theoretic, linear-algebraic framework for Tate modules of abelian varieties under isogeny by packaging torsion into the adelic module and its rationalization , with Galois action given by . The central contribution is a functor from the isogeny category to sublattices of , yielding an equivalence of categories and a natural identification that translates isogenies into lattice inclusions; this provides a computable dictionary between isogenous abelian varieties and -stable lattices. The framework enables explicit change-of-basis computations for Galois representations and polarizations, illustrated by an extended example showing how to determine -adic actions on and on via polarization data. Together, these results furnish a structured toolkit for analyzing Galois images of torsion and Tate modules in arithmetic geometry, with practical matrix-calculation methods for isogeny-related questions.

Abstract

We explain the linear algebraic framework provided by Tate modules of isogenous abelian varieties in a category-theoretic way.

Paper Structure

This paper contains 19 sections, 11 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Results
  4. Contents
  5. Background on abelian varieties
  6. Isogenies and Tate modules
  7. Duals
  8. Weil pairing
  9. Polarizations
  10. Functorial aspects of Tate modules
  11. A guiding example
  12. From isogenies to Tate modules
  13. Change of basis
  14. Bases
  15. Isogenies given in terms of kernels
  16. ...and 4 more sections

Key Result

Theorem 1.2.1

The association $\varphi \mapsto \Lambda\varphi$ determines a functor $\Lambda$ from to The functor $\Lambda$ has the following properties.

Theorems & Definitions (39)

  • Theorem 1.2.1
  • Proposition 2.1.4
  • proof
  • Example 2.1.8
  • Remark 2.1.11
  • Proposition 2.1.12
  • proof
  • Example 2.2.5
  • Example 2.2.9
  • Theorem 2.3.1: Tautological Weil pairing
  • ...and 29 more