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].

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

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Summary

Introduction

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

Background on Galois representations
Stable lattices and the Bruhat–Tits tree
Characters and quadratic extensions
Identifying quadratic extensions
Determining the residual representation
Identifying cubic extensions
Determining residual irreducibility and splitting field
Determining triviality of the residual representation up to isogeny
The test function for small isogeny classes
Quadratically independent sets of primes
Large isogeny classes
Determining the four characters: the test
Detecting triviality of the semisimplification
Further examples
Full Text
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