This package provides functionality for working with Galois representations
Gal(bar(K)/K) -> GLm(C)
over p-adic fields K (type FldPad). The representations we consider are precisely those on the `Galois side' of the local Langlands correspondence, or, technically, the Frobenius-semisimple Weil-Deligne representations over K. We refer the reader to Tate's article [Tat79, Para 4] for their basic properties.
In arithmetic geometry, such representations arise from l-adic étale cohomology of algebraic varieties over K for l≠p. Magma includes Galois representations that come from finite Galois groups of p-adic extensions, local components of Dirichlet characters and Artin representations, local Galois representations coming from a Tate module of an elliptic curve or a modular form, as well as various constructions to produce new representations from existing ones, such as direct sums, tensor products, induction, restriction and semisimplification.
Suppose K is a finite extension of Qp, with ring of integers O, maximal ideal m and residue field k=O/m isomorphic to Fq. The absolute Galois group of K fits into an exact sequence 1 -> I_K -> Gal(bar{K}/K) -> Gal(bar k/k) -> 1, σ |-> σmod m where IK is the inertia group of K. The group Gal(bar k/k) is topologically generated by the automorphism x |-> xq, and any of its lifts to Gal(bar(K)/K) is called an arithmetic Frobenius element, denoted FrobK.
The Weil group WK of K is a subgroup of Gal(bar(K)/K) generated by the inertia group IK and any Frobenius element. It fits in the same exact sequence as above, except that the profinite group Gal(bar k/k) isomorphic to hatZ is replaced by a copy of Z with discrete topology. A Weil representation is a continuous representation
ρ: WK -> GLm(C).
By continuity, the image of inertia ρ(IK) is finite, and ρ is said to be unramified if it is trivial. We will always assume that ρ is Frobenius-semisimple, that is ρ(FrobK) is a semisimple endomorphism of Cm. Every such representation ρ is a direct sum of irreducible representations of the form
ψ tensor R,
where ψ: IK |-> 1, FrobK |-> α∈C x is an unramified 1-dimensional character (uniquely determined by α∈C x ) and R is a representation of Gal(F/K) for some finite Galois extension F/K.
Weil representations occur naturally as local components of Artin representations, and as representations associated to (H1 of) elliptic curves and abelian varieties over K with potentially good reduction. (By the Néron-Ogg-Shafarevich criterion, potentially good reduction is equivalent to ρ(IK) being finite.) To deal with arbitrary reduction behaviour one considers, more generally, Weil-Deligne representations. These come from Galois representations
ρ: Gal(bar(K)/K) -> GLm(C)
that can be described as follows.
As before, ρ is a direct sum of indecomposable representations, and an indecomposable one is now of the form (cf. [Tat79, 4.1.5])
ρ = ψ tensor SP(n) tensor R,
with ψ and R as above, and SP(n) is a `special' representation ([Tat79, 4.1.4]),
SP(n) = C e0 + C e1 + ... + C en - 1.
The ei are eigenvectors for FrobK with eigenvalues q - i, and the action of inertia is nilpotent, and represented by a matrix N (see [Tat79, 4.1.2]) that takes ei |-> ei + 1 and en - 1 |-> 0.
This is precisely how Galois representations are stored in Magma,
ρ = bigoplusi ψi tensor SP(ni) tensor Ri,
except that it is convenient to
(1) assume that all Ri factor through the same Galois group G=Gal(F/K), and
(2) allow ψi to be arbitrary unramified representations, not necessarily 1-dimensional. Such a ψ is a sum of unramified characters χj: FrobK |-> αj, and ψ is uniquely determined by its Euler factor
Ψ(T) = det (1 - FrobK - 1T|ψ) = ∏j (1 - αj - 1 T) ∈C[T].
In other words, every component of ρ can be represented (though not quite uniquely) by a triple < Ψ, n, c >, where Ψ∈C[T], n∈Z and c is a character of a representation R of a finite group Gal(F/K). Throughout the chapter we will refer to R as the `finite part' of an indecomposable representation ψ tensor SP(n) tensor R, though it depends on the choice of such a presentation.
Galois representations have Magma type GalRep. The base field K, field F, the Galois group F/K, and the list of components of ρ can be obtained with BaseField, Field, Group and Factorization, respectively.
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(Rationals());
First, construct a typical unramified representation specified by its Euler factor ψ.
> psi:=UnramifiedRepresentation(K,1+x^2); psi; 2-dim unramified Galois representation Unr(1+x^2) over Q2[20]
Next, construct a typical finite image representation, by picking a character of a finite Galois extension F/K, in this case an S3-extension.
> S<z>:=PolynomialRing(K); > F:=ext<K|z^3-2>; > c:=GaloisRepresentations(F,K)[3]; c; 2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[20]
Here is a general Galois representation, the tensor product of all 3 terms, and another one --- direct sum of the same three terms.
> psi*SP(K,2)*c; 8-dim Galois representation Unr(1+x^2)*SP(2)*(2,0,-1) with G=S3, I=C3, conductor 2^8 over Q2[20] > psi+SP(K,2)+c; 6-dim Galois representation Unr(1+x^2) + (2,0,-1) + SP(2) with G=S3, I=C3, conductor 2^3 over Q2[20]
There are various choices of signs in local class field theory, for which there appears to be no consensus in the literature. Our conventions follow Tate [Tat79], and are as follows. (We refer the reader to Tate's article [Tat79] for the definitions and properties of Weil-Deligne representations, and [Tat79], [Del79] for their ε-factors.)
For a p-adic field K with ring of integers O, uniformizer π and residue field Fq,
A Frobenius element FrobK is an arithmetic Frobenius, i.e. acts as x |-> xq on the residue field (not x |-> x1/q).
The local reciprocity map θ: K x to Gal(bar K/K)ab takes π to FrobK - 1 (not FrobK).
The local epsilon-factors ε(χ)=ε(χ, ψ, dx) rely implicitly on the choice of a measure dx on O and an additive character ψ: K to C. Our choices are that dx is normalized, intO dx=1, and
ψ(x) = exp(2π i TrK/Qp(x)),
viewing TrK/Qp(x)∈Qp as any rational number a/pn∈Q in the same class mod Zp. Finally, for 1-dimensional ramified χ, the formula for ε(χ) is as in [Tat79, 3.2.6.2],
ε(χ) = int_(c - 1 O x ) χ - 1(θ(x)) ψ(x),
with
vK(c) = vK((Conductor)(χ)) + vp((Discriminant)(O, Zp)).
Euler factors of global Artin representations A (and of Dirichlet characters) are the same (not complex conjugate) as the local ones:
hfilEulerFactor(A,p) = EulerFactor(GaloisRepresentation(A,p)).hfil
As the Artin L-function L(A, s) is defined using arithmetic Frobenius and Galois representations using geometric Frobenius, this means that the Galois representation GaloisRepresentation(A,p) comes from the Galois action on the dual vector space of A.
Galois representations attached to Artin representations are computed using the machinery of [DD13]. Galois representations coming from elliptic curves rely partly on the theory of reconstructing representations from their Euler factors [DD15] (see ParaExample: Reconstructing a Galois Representation from its Euler Factors), and Rachel Newton's tame local reciprocity formula [New12].