# Introduction

This package enables one to do the following. Starting with a cuspidal newform, one may define the local component at $p$ of the associated automorphic representation, and determine its key properties. Furthermore, via the local Langlands correspondence, there exists a related Galois representation on the absolute Galois group of ${\mathbb{Q}}_p.$ One may compute that Galois representation (completely in many cases; unsupported cases raise an error).

The algorithms implemented here are described in [[Loeffler and Weinstein, n.d.](../../references.md#cite-loeffler-weinstein)].

## Motivation

Let $F$ be a local non-archimedean field and let $G$ be a reductive group over $F$. The representation theory of $G$ is a rich subject in its own right but also has fascinating (and often conjectural) connections with the representation theory of the absolute Galois group of $F$. This package deals with admissible irreducible representations in the case that $G$ is the group ${\operatorname{GL}}_2$ and $F$ is the $p$-adic field ${\mathbb{Q}}_p$. Such objects correspond canonically to two-dimensional representations of the absolute Galois group of $F$ of a certain sort.

There are applications to the study of local properties of (global) Galois representations arising from modular forms. Namely, if $f$ is a cuspidal newform, then there is associated to $f$ a family of $\ell$-adic Galois representations $\rho_f\colon{\operatorname{Gal}}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})\to{\operatorname{GL}}_2(\overline{{\mathbb{Q}}}_\ell)$. For example, for a prime $p$ different from $\ell$, one may determine the restriction of $\rho_f$ to the decomposition group at $p$.

## Definitions

Here we introduce the category of admissible irreducible representations of ${\operatorname{GL}}_2$ over a non-archimedean field, which for our purposes will be ${\mathbb{Q}}_p$. The first systematic study of admissible representations is [[Jacquet and Langlands, 1970](../../references.md#cite-jacquetlanglands)]. For an accessible introduction, see [[Bushnell and Henniart, 2006](../../references.md#cite-bushnellhenniart)].

Let $G$ be the locally compact group ${\operatorname{GL}}_2({\mathbb{Q}}_p)$. An *admissible* representation of $G$ on a complex vector space $V$ is a homomorphism $\pi\colon G\to {\operatorname{Aut}}V$ which satisfies the properties:

**(i)**
every vector $v\in V$ is fixed by a compact open subgroup of $G$, and

**(ii)**
for every compact open subgroup $K\subset G$, $V^K$ is finite-dimensional.

The center of $G$ is ${\mathbb{Q}}_p^\times$. If $\pi$ is an irreducible admissible representation of $G$, then it has a unique *central character* $\varepsilon\colon{\mathbb{Q}}_p^\times\to{\mathbb{C}}^\times$ such that $\pi(g)$ acts as the scalar $\varepsilon(g)$ for all $g\in {\mathbb{Q}}_p^\times$. The *conductor* of an irreducible admissible representation $\pi$ is a measure of how small a compact open subgroup $K\subset G$ must be before one sees nonzero $K$-invariant vectors; see [[Casselman, 1973](../../references.md#cite-casselman)]. Consider the filtration $K_0(p^n)$ of subgroups of $G$, where $K_0(p^n)$ is the subgroup of matrices $\begin{pmatrix}a & b  \\ c & d\end{pmatrix} \in {\operatorname{GL}}_2({\mathbb{Z}}_p)$ with $c\equiv 0\pmod{p^n}$. If $\pi$ admits a nonzero vector fixed by $K_0(1)={\operatorname{GL}}_2({\mathbb{Z}}_p)$, then $\pi$ is called *spherical* or *unramified principal series* and has conductor 1. If $\pi$ admits a nonzero vector $v$ for which $\pi\left( \begin{pmatrix}a & b  \\ c & d\end{pmatrix} \right)v=\varepsilon(a)v$ for all $\begin{pmatrix}a & b  \\ c & d\end{pmatrix} \in K_0(p^n)$, and $n\geq 1$ is minimal for this condition, then the conductor of $\pi$ is $p^n$. (Note the similarity with the convention used to define the level of a modular form for $\Gamma_0(N)$.) In both cases the vector $v$ so described is unique up to scaling; see [[Casselman, 1973](../../references.md#cite-casselman)]. We shall call $v$ a *new vector* for $\pi$.

If $\chi$ is a character of ${\mathbb{Q}}_p^\times$ then let $\pi\otimes\chi$ be the representation $g\mapsto \chi(g)\pi(g)$; such a representation is a *twist* of $\chi$. If $\pi$ has minimal conductor among all its twists $\pi\otimes\chi$, then $\pi$ is called *minimal*.

Admissible representations are generally infinite-dimensional, but we will nonetheless be able to present them using Magma infrastructure for representations of finite groups and Dirichlet characters.

## The Principal Series

We can directly construct a large class of admissible representations of $G$. Let $\chi_1$ and $\chi_2$ be two characters of ${\mathbb{Q}}_p^*$. Let $B\subset G$ be the Borel subgroup of upper triangular matrices. Then $\begin{pmatrix}a & b  \\ 0 & d\end{pmatrix} \mapsto |a/d|^{-1/2}\chi_1(a)\chi_2(d)$ is a character $\chi$ of $B$. An admissible representation $\pi$ is a *principal series* representation if it is a composition factor of the induced representation $\pi(\chi_1,\chi_2):={\operatorname{Ind}}_B^G \chi$. This induced representation is already irreducible unless $\chi_1\chi_2^{-1}$ equals $|.|^{\pm 1}$, in which case it has length two, with one 1-dimensional and one infinite-dimensional composition factor. For instance, ${\operatorname{Ind}}_B^G 1$ has a trivial 1-dimensional submodule and an irreducible infinite-dimensional quotient ${\operatorname{St}}_G$, the *Steinberg* representation. The unramified principal series representations are either 1-dimensional, in which case they factor through the determinant map, or else they take the form $\pi(\chi_1,\chi_2)$, where $\chi_1$ and $\chi_2$ are unramified characters of ${\mathbb{Q}}_p^\times$ (meaning they are trivial on ${\mathbb{Z}}_p^\times$).

The central character of $\pi(\chi_1,\chi_2)$ is $\chi_1\chi_2$, and its conductor is the product of the conductors of the $\chi_i$. Note that $\pi(\chi_1,\chi_2)$ is minimal if and only if one of the characters $\chi_1,\chi_2$ is unramified. The Steinberg representation ${\operatorname{St}}_G$ has trivial central character and conductor $p$.

## Supercuspidal Representations

For the purposes of this package, an admissible irreducible representation is *supercuspidal* if it does not belong to the principal series. A supercuspidal representation $\pi$ has conductor $p^c$, where $c\geq 2$. There is a convenient, if technical, classification of supercuspidal representations of $G$. Let $\pi$ be supercuspidal. By [[Bushnell and Henniart, 2006](../../references.md#cite-bushnellhenniart)], Ch. 15, there is a representation $\Xi$ of an open and compact-mod-center subgroup $K\subset G$ for which $\pi={\operatorname{Ind}}_K^G\Xi$. If $c$ is even, then we may take $K$ to be ${\mathbb{Q}}_p^\times{\operatorname{GL}}_2({\mathbb{Z}}_p)$, and if $c$ is odd, we may take $K$ to be the normalizer of the Iwahori subgroup $K_0(p) = \begin{pmatrix}{\mathbb{Z}}_p^\times & {\mathbb{Z}}_p \\ p{\mathbb{Z}}_p & {\mathbb{Z}}_p^\times\end{pmatrix}$ in $G$. We call the pair $(K,\Xi)$ a *cuspidal inducing datum*.

## The Local Langlands Correspondence

The Local Langlands Correspondence is a canonical bijection $\pi\mapsto\sigma(\pi)$ between irreducible admissible representations of $G$ and local 2-dimensional representations of the absolute Galois group of ${\mathbb{Q}}_p$ of a certain sort. (Note to purists: the proper Galois-theoretic object to study in this scenario is the Weil-Deligne representation, which consists of the datum of a representation of the Weil group, together with a monodromy operator. See [[Tate, 1979](../../references.md#cite-tate)].) The bijection manifests as an agreement of $L$- and $\varepsilon$-factors that one constructs for each category. The foundation for the Local Langlands Correspondence for ${\operatorname{GL}}_2$ over a non-archimedean field was laid in [[Jacquet and Langlands, 1970](../../references.md#cite-jacquetlanglands)]; the work was completed in [[Kutzko, 1980](../../references.md#cite-kutzko1)] and [[Kutzko, 1984](../../references.md#cite-kutzko2)].

We remark that the conductor of $\pi$ agrees with the Artin conductor of $\sigma(\pi)$, and that $\pi$ is principal series (resp., Steinberg, supercuspidal) if and only if $\sigma(\pi)$ is a sum of two characters (resp., reducible but not decomposable, irreducible). We also remark that if $p\neq 2$ and $\sigma$ is an irreducible 2-dimensional Galois representation of ${\mathbb{Q}}_p$, then $\sigma$ must be induced from a character $\chi$ of a quadratic field extension $E/{\mathbb{Q}}_p$. Then $(E,\chi)$ is called an *admissible pair* (see [[Bushnell and Henniart, 2006](../../references.md#cite-bushnellhenniart)], Ch. 18).

## Connection with Modular Forms

The classical theory of modular forms has a modern interpretation in terms of *cuspidal automorphic representations*. These are representations $\Pi$ of the adele group ${\operatorname{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ which appear in the Hilbert space of square-integrable cuspidal functions $L^2_0({\operatorname{GL}}_2({\mathbb{Q}})\backslash{\operatorname{GL}}_2({\mathbb{A}}_{\mathbb{Q}}),\varepsilon)$, where $\varepsilon$ is a Dirichlet character. Let $f$ be a cuspidal newform for $\Gamma_0(N)$ with Dirichlet character $\varepsilon$. Then there is associated to $f$ a cuspidal automorphic representation $\Pi_f$, see [[Gelbart, 1975](../../references.md#cite-gelbart)]. This is a restricted tensor product $\bigotimes_{p\leq\infty} \pi_{f,p}$, where if $p$ is a finite prime, $\pi_{f,p}$ is an admissible representation of ${\operatorname{GL}}_2({\mathbb{Q}}_p)$. There is also the Galois representation $\rho_f$ attached to $f$ constructed by Deligne. By [[Carayol, 1983](../../references.md#cite-carayol)] there is a straightforward relationship between $\sigma(\pi_{f,p})$ and the restriction of $\rho_f$ to the decomposition group at $p$. Therefore to determine the local properties of $\rho_f$ it is enough to compute the local components $\pi_{f,p}$. These are almost always unramified principal series; the only challenge is to compute $\pi_{f,p}$ when $p$ divides $N$.

## Category

In Magma, admissible representations are objects of type `RepLoc`.

## Verbose Output

To see information about computations in progress, enter `SetVerbose("RepLoc", 1)`.