Black Box Galois Representations

TL;DR

This paper introduces methods to analyze 2-dimensional 2-adic Galois representations from a black box perspective, determining key properties using only characteristic polynomials at select primes, with applications to number fields and Bianchi modular forms.

Contribution

It develops a novel approach to extract detailed information about Galois representations from limited data, including determinant and reducibility, using finite test sets of primes.

Findings

Determines the determinant of the Galois representation.

Identifies whether the representation is residually reducible.

Provides insights into the isogeny graph size of the representation.

Abstract

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\det\rho$, whether or not $\rho$ is residually reducible, and further information about the size of the isogeny graph of $\rho$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $\rho$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.