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 detρ, 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].
Highlights
Modularity of elliptic curves over K can be interpreted as a statement that the 2-dimensional Galois representation arising from the action of GK on the -adic Tate module of the elliptic curve is equivalent, as a representation, to a representation attached to a suitable automorphic form over K
In this 2-dimensional context and with = 2, techniques have been developed by Serre [15], Faltings, Livné [13] and others to establish such an equivalence using only the characteristic polynomial of ρ(σ) for a finite number of elements σ ∈ GK
The ramified set of primes S is known in advance and the Galois automorphisms σ which are used in the Serre–Faltings–Livné method have the form σ = Frob p where p is a prime not in S, so that ρ is unramified at p
Summary
In this paper we study Galois representations of K as “Black Boxes” where both the base field K and the finite ramified set S are specified in advance, and the only information we have about ρ is the characteristic polynomial of ρ(Frob p) for certain primes p not in S; we may specify these primes, but only finitely many of them Using such a Black Box as an oracle, we seek to give algorithmic answers to questions such as the following (see the following section for definitions):. For example, we can determine ρ (mod 4) when ρ is trivial, and as a final application, in Section 7 we give a (finite) criterion for whether ρ has trivial semisimplification For each of these tasks we will define a finite set T of primes of K, disjoint from S, such that the Black Box information about ρ(Frob p) for p ∈ T is sufficient to answer the question under consideration. This includes general-purpose code for computing the test sets T0, T1 and T2 from a number field K and a set S of primes of K, and worked examples which reproduce the examples we give in the text
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