Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Comparing Galois representations in the residually reducible case

Nuno Freitas, Ignasi Sánchez-Rodríguez

TL;DR

The paper develops and implements a practical variant of Grenié’s theorem to decide when two $p$-adic Galois representations with residually reducible images have isomorphic semisimplifications, by constructing a smaller extension $K_S$ and testing Frobenius data on a finite set of primes. It leverages Selmer-group techniques and a controlled pro-$p$ extension framework to replace large-scale field computations with a focused, algorithmic approach that is robust in the residually reducible setting. The authors demonstrate the method on a $3$-adic example related to Grenié’s work, recover Grenié’s result in an automated way, and apply it to modularity questions for abelian surfaces, including concrete examples with conductors $3^7$ and $3^{10}$, using either characteristic polynomials or traces at carefully chosen primes. They also introduce a stopping criterion to avoid GRH in many computations, increasing practicality. Overall, the work expands the applicability of the Faltings–Serre framework to residually reducible representations and provides tools for automated verification of modularity statements in low-dimensional cases via $3$-adic data.

Abstract

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $ρ_1, ρ_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of $ρ_1$ and $ρ_2$ are isomorphic. As an application, we show that an irreducible representation $ρ: G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié's result in a fully automated way.

Comparing Galois representations in the residually reducible case

TL;DR

The paper develops and implements a practical variant of Grenié’s theorem to decide when two -adic Galois representations with residually reducible images have isomorphic semisimplifications, by constructing a smaller extension and testing Frobenius data on a finite set of primes. It leverages Selmer-group techniques and a controlled pro- extension framework to replace large-scale field computations with a focused, algorithmic approach that is robust in the residually reducible setting. The authors demonstrate the method on a -adic example related to Grenié’s work, recover Grenié’s result in an automated way, and apply it to modularity questions for abelian surfaces, including concrete examples with conductors and , using either characteristic polynomials or traces at carefully chosen primes. They also introduce a stopping criterion to avoid GRH in many computations, increasing practicality. Overall, the work expands the applicability of the Faltings–Serre framework to residually reducible representations and provides tools for automated verification of modularity statements in low-dimensional cases via -adic data.

Abstract

Let and be a prime. Let be a number field and consider two Galois representations having residual image a -group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of and are isomorphic. As an application, we show that an irreducible representation unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié's result in a fully automated way.
Paper Structure (12 sections, 14 theorems, 25 equations)

This paper contains 12 sections, 14 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Computational resources
  3. Acknowledgements
  4. A theorem of Grenié
  5. Auxiliary results
  6. A smaller extension
  7. The construction of
  8. The example of Grenié
  9. Modularity of abelian surfaces via 3-adic representations
  10. A surface of conductor $3^7$
  11. A surface of conductor $3^{10}$
  12. Avoiding GRH

Key Result

Corollary 1.1

Let $\rho_1, \rho_2 : G_K \to \operatorname{GL}_2(\mathbb{Z}_3)$ be continuous representations unramified outside $\mathfrak{q}_3$ having the same determinant. Suppose their determinant is trivial modulo $3$ and consider the list of primes Then $\rho_1$ and $\rho_2$ have isomorphic semisimplifications if and only if $\rho_1(\operatorname{Frob}_t)$ and $\rho_2(\operatorname{Frob}_t)$ have the same

Theorems & Definitions (37)

  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2.1: Grenié
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 27 more