Overview#
This package provides functionality for working with Galois representations
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 [Tate, 1979, §4] for their basic properties.
In arithmetic geometry, such representations arise from \(l\)-adic étale cohomology of algebraic varieties over \(K\) for \(l\ne p\). 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.
Notation and Printing#
Suppose \(K\) is a finite extension of \({\mathbb{Q}}_p\), with ring of integers \(O\), maximal ideal \(m\) and residue field \(k=O/m\cong{\mathbb{F}}_q\). The absolute Galois group of \(K\) fits into an exact sequence
where \(I_K\) is the inertia group of \(K\). The group \(\mathop{\rm Gal}\nolimits(\bar k/k)\) is topologically generated by the automorphism \(x\mapsto x^q\), and any of its lifts to \(\mathop{\rm Gal}\nolimits(\bar{K}/K)\) is called an arithmetic Frobenius element, denoted \(\mathop{\rm Frob}\nolimits_K\).
The Weil group \(W_K\) of \(K\) is a subgroup of \(\mathop{\rm Gal}\nolimits(\bar{K}/K)\) generated by the inertia group \(I_K\) and any Frobenius element. It fits in the same exact sequence as above, except that the profinite group \(\mathop{\rm Gal}\nolimits(\bar k/k)\cong\hat{\mathbb{Z}}\) is replaced by a copy of \({\mathbb{Z}}\) with discrete topology. A Weil representation is a continuous representation
By continuity, the image of inertia \(\rho(I_K)\) is finite, and \(\rho\) is said to be unramified if it is trivial. We will always assume that \(\rho\) is Frobenius-semisimple, that is \(\rho(\mathop{\rm Frob}\nolimits_K)\) is a semisimple endomorphism of \({\mathbb{C}}^m\). Every such representation \(\rho\) is a direct sum of irreducible representations of the form
where \(\psi: I_K \mapsto 1, \mathop{\rm Frob}\nolimits_K\mapsto \alpha\in{\mathbb{C}}^\times\) is an unramified 1-dimensional character (uniquely determined by \(\alpha\in{\mathbb{C}}^\times\)) and \(R\) is a representation of \(\mathop{\rm Gal}\nolimits(F/K)\) for some finite Galois extension \(F/K\).
Weil representations occur naturally as local components of Artin representations, and as representations associated to (\(H^1\) of) elliptic curves and abelian varieties over \(K\) with potentially good reduction. (By the Néron-Ogg-Shafarevich criterion, potentially good reduction is equivalent to \(\rho(I_K)\) being finite.) To deal with arbitrary reduction behaviour one considers, more generally, Weil-Deligne representations. These come from Galois representations
that can be described as follows.
As before, \(\rho\) is a direct sum of indecomposable representations, and an indecomposable one is now of the form (cf. [Tate, 1979, 4.1.5])
with \(\psi\) and \(R\) as above, and \(\hbox{\tt SP}(n)\) is a ‘special’ representation ([Tate, 1979, 4.1.4]),
The \(e_i\) are eigenvectors for \(\mathop{\rm Frob}\nolimits_K\) with eigenvalues \(q^{-i}\), and the action of inertia is nilpotent, and represented by a matrix \(N\) (see [Tate, 1979, 4.1.2]) that takes \(e_i\mapsto e_{i+1}\) and \(e_{n-1}\mapsto 0\).
This is precisely how Galois representations are stored in Magma,
except that it is convenient to
(1) assume that all \(R_i\) factor through the same Galois group \(G=\mathop{\rm Gal}\nolimits(F/K)\), and
(2) allow \(\psi_i\) to be arbitrary unramified representations, not necessarily 1-dimensional. Such a \(\psi\) is a sum of unramified characters \(\chi_j: \mathop{\rm Frob}\nolimits_K\mapsto \alpha_j\), and \(\psi\) is uniquely determined by its Euler factor
In other words, every component of \(\rho\) can be represented (though not quite uniquely) by a triple \(\langle \Psi,n,c \rangle\), where \(\Psi\in {\mathbb{C}}[T]\), \(n\in {\mathbb{Z}}\) and \(c\) is a character of a representation \(R\) of a finite group \(\mathop{\rm Gal}\nolimits(F/K)\). Throughout the chapter we will refer to \(R\) as the ‘finite part’ of an indecomposable representation \(\psi\otimes\hbox{\tt SP}(n)\otimes R\), 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 \(\rho\) can be obtained with BaseField, Field, Group and Factorization, respectively.
- Example: Galrep Printing (ex-8ff07a)#
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(Rationals());
First, construct a typical unramified representation specified by its Euler factor \(\psi\).
> 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 \(S_3\)-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]
Conventions#
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 [Tate, 1979], and are as follows. (We refer the reader to Tate’s article [Tate, 1979] for the definitions and properties of Weil-Deligne representations, and [Deligne, 1979, Tate, 1979] for their \(\epsilon\)-factors.)
For a \(p\)-adic field \(K\) with ring of integers \(O\), uniformizer \(\pi\) and residue field \({\mathbb{F}}_q\),
\(\bullet\,\,\,\) A Frobenius element \(\mathop{\rm Frob}\nolimits_K\) is an arithmetic Frobenius, i.e. acts as \(x\mapsto x^q\) on the residue field (not \(x\mapsto x^{1/q}\)).
\(\bullet\,\,\,\) The local reciprocity map \(\theta: K^\times\to \mathop{\rm Gal}\nolimits(\bar K/K)^{ab}\) takes \(\pi\) to \(\mathop{\rm Frob}\nolimits_K^{-1}\) (not \(\mathop{\rm Frob}\nolimits_K\)).
\(\bullet\,\,\,\) The local epsilon-factors \(\epsilon(\chi)=\epsilon(\chi,\psi,dx)\) rely implicitly on the choice of a measure \(dx\) on \(O\) and an additive character \(\psi: K\to {\mathbb{C}}\). Our choices are that \(dx\) is normalized, \(\int_O dx=1\), and
viewing \(\mathop{\rm Tr}\nolimits_{K/{\mathbb{Q}}_p}(x)\in{\mathbb{Q}}_p\) as any rational number \(a/p^n\in{\mathbb{Q}}\) in the same class mod \({\mathbb{Z}}_p\). Finally, for 1-dimensional ramified \(\chi\), the formula for \(\epsilon(\chi)\) is as in [Tate, 1979, 3.2.6.2],
with
\(\bullet\,\,\,\) Euler factors of global Artin representations \(A\) (and of Dirichlet characters) are the same (not complex conjugate) as the local ones:
EulerFactor(A,p) = EulerFactor(GaloisRepresentation(A,p)).
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 \(A\).
Implementation Notes#
Galois representations attached to Artin representations are computed using the machinery of [Dokchitser and Dokchitser, 2013]. Galois representations coming from elliptic curves rely partly on the theory of reconstructing representations from their Euler factors [Dokchitser and Dokchitser, 2015] (see §Example: Reconstructing a Galois Representation from its Euler Factors), and Rachel Newton’s tame local reciprocity formula [Newton, 2012].