Black Box Galois representations

Abstract

We develop methods to study 2-dimensional 2-adic Galois representations ρ 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 ρ, whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K = Q, and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form.These results form part of the first author's thesis [2].arXiv:1709.06430v5 [math.NT]