# Overview

This package provides functionality for working with Galois representations

$$
\mathop{\rm Gal}\nolimits(\bar{K}/K) \longrightarrow \mathop{\rm GL}\nolimits_m({\mathbb{C}})
$$

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 [[Tate, 1979](../../references.md#cite-tate), §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

$$
\begin{matrix}1 &\longrightarrow& I_K &\longrightarrow& \mathop{\rm Gal}\nolimits(\bar{K}/K) &\longrightarrow& \mathop{\rm Gal}\nolimits(\bar k/k) &\longrightarrow& 1, \\
    &    &     &    &    \sigma      &\mapsto& \sigma\ \mathop{\rm mod}\ m\end{matrix}
$$

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

$$
\rho: W_K \longrightarrow \mathop{\rm GL}\nolimits_m({\mathbb{C}}).
$$

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

$$
\psi \otimes R,
$$

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

$$
\rho: \mathop{\rm Gal}\nolimits(\bar{K}/K) \longrightarrow \mathop{\rm GL}\nolimits_m({\mathbb{C}})
$$

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](../../references.md#cite-tate), 4.1.5])

$$
\rho = \psi \otimes\hbox{\tt SP}(n) \otimes R,
$$

with $\psi$ and $R$ as above, and $\hbox{\tt SP}(n)$ is a ‘special’ representation ([[Tate, 1979](../../references.md#cite-tate), 4.1.4]),

$$
\hbox{\tt SP}(n) = {\mathbb{C}}e_0 + {\mathbb{C}}e_1 + \ldots + {\mathbb{C}}e_{n-1}.
$$

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](../../references.md#cite-tate), 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,

$$
\rho = \bigoplus_i\,\, \psi_i \otimes\hbox{\tt SP}(n_i) \otimes R_i,
$$

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*

$$
\Psi(T) = \det (1-\mathop{\rm Frob}\nolimits_K^{-1}T|\psi) = \prod_j (1-\alpha_j^{-1} T) \in {\mathbb{C}}[T].
$$

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)`

```magma
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(Rationals());

```

First, construct a typical unramified representation specified by its Euler factor $\psi$.

```magma
> 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.

```magma
> 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.

```magma
> 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](../../references.md#cite-tate)], and are as follows. (We refer the reader to Tate’s article [[Tate, 1979](../../references.md#cite-tate)] for the definitions and properties of Weil-Deligne representations, and [[Deligne, 1979](../../references.md#cite-deligne-rootnumber), [Tate, 1979](../../references.md#cite-tate)] 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

$$
\psi(x) = \exp\bigl(2\pi i\,\mathop{\rm Tr}\nolimits_{K/{\mathbb{Q}}_p}(x)\bigr),
$$

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](../../references.md#cite-tate), 3.2.6.2],

$$
\epsilon(\chi) = \int_{c^{-1} O^\times} \chi^{-1}(\theta(x)) \psi(x),
$$

with

$$
v_K(c)\,=\,v_K({\rm Conductor}(\chi)) + v_p({\rm Discriminant}(O,{\mathbb{Z}}_p)).
$$

$\bullet\,\,\,$ 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](../../references.md#cite-ddfrob13)]. Galois representations coming from elliptic curves rely partly on the theory of reconstructing representations from their Euler factors [[Dokchitser and Dokchitser, 2015](../../references.md#cite-ddweil15)] (see §[Example: Reconstructing a Galois Representation from its Euler Factors](fields-2.md#galrep-ex2)), and Rachel Newton’s tame local reciprocity formula [[Newton, 2012](../../references.md#cite-newton2012)].