Introduction
============

This package enables one to do the following. Starting with a cuspidal newform,
one may define the local component at :math:`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 :math:`{\mathbb{Q}}_p.` One may compute (the
restriction to inertia of) that Galois representation.

The algorithms implemented here are described in
:cite:`loeffler-weinstein`.

Motivation
----------

Let :math:`F` be a local non-archimedean field and let :math:`G` be a reductive
group over :math:`F`. The representation theory of :math:`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 :math:`F`. This
package deals with admissible irreducible representations in the case that
:math:`G` is the group :math:`{\operatorname{GL}}_2` and :math:`F` is the
:math:`p`-adic field :math:`{\mathbb{Q}}_p`. Such objects correspond canonically
to two-dimensional representations of the absolute Galois group of :math:`F` of
a certain sort.

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

Definitions
-----------

Here we introduce the category of admissible irreducible representations of
:math:`{\operatorname{GL}}_2` over a non-archimedean field, which for our
purposes will be :math:`{\mathbb{Q}}_p`. The first systematic study of
admissible representations is :cite:`JacquetLanglands`. For an
accessible introduction, see :cite:`BushnellHenniart`.

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

(#) every vector :math:`v\in V` is fixed by a compact open subgroup of
    :math:`G`, and

(#) for every compact open subgroup :math:`K\subset G`, :math:`V^K` is
    finite-dimensional.

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

If :math:`\chi` is a character of :math:`{\mathbb{Q}}_p^\times` then let
:math:`\pi\otimes\chi` be the representation :math:`g\mapsto \chi(g)\pi(g)`;
such a representation is a *twist* of :math:`\chi`. If :math:`\pi` has minimal
conductor among all its twists :math:`\pi\otimes\chi`, then :math:`\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
:math:`G`. Let :math:`\chi_1` and :math:`\chi_2` be two characters of
:math:`{\mathbb{Q}}_p^*`. Let :math:`B\subset G` be the Borel subgroup of upper
triangular matrices. Then
:math:`\begin{pmatrix} a & b  \\ 0 & d \end{pmatrix} \mapsto |a/d|^{-1/2}\chi_1(a)\chi_2(d)` is a
character :math:`\chi` of :math:`B`. An admissible representation :math:`\pi` is
a *principal series* representation if it is a composition factor of the induced
representation :math:`\pi(\chi_1,\chi_2):={\operatorname{Ind}}_B^G \chi`. This
induced representation is already irreducible unless :math:`\chi_1\chi_2^{-1}`
equals :math:`|.|^{\pm 1}`, in which case it has length two, with one
1-dimensional and one infinite-dimensional composition factor. For instance,
:math:`{\operatorname{Ind}}_B^G 1` has a trivial 1-dimensional submodule and an
irreducible infinite-dimensional quotient :math:`{\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 :math:`\pi(\chi_1,\chi_2)`, where :math:`\chi_1` and
:math:`\chi_2` are unramified characters of :math:`{\mathbb{Q}}_p^\times`
(meaning they are trivial on :math:`{\mathbb{Z}}_p^\times`).

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

.. _sec-supercuspidal:

.. _supercuspidal:

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 :math:`\pi` has conductor :math:`p^c`, where :math:`c\geq 2`.
There is a convenient, if technical, classification of supercuspidal
representations of :math:`G`. Let :math:`\pi` be supercuspidal.
By :cite:`BushnellHenniart`, Ch. 15, there is a representation
:math:`\Xi` of an open and compact-mod-center subgroup :math:`K\subset G` for
which :math:`\pi={\operatorname{Ind}}_K^G\Xi`. If :math:`c` is even, then we may
take :math:`K` to be
:math:`{\mathbb{Q}}_p^\times{\operatorname{GL}}_2({\mathbb{Z}}_p)`, and if
:math:`c` is odd, we may take :math:`K` to be the normalizer of the Iwahori
subgroup
:math:`K_0(p) = \begin{pmatrix} {\mathbb{Z}}_p^\times & {\mathbb{Z}}_p \\ p{\mathbb{Z}}_p & {\mathbb{Z}}_p^\times \end{pmatrix}`
in :math:`G`. We call the pair :math:`(K,\Xi)` a *cuspidal inducing datum*.

.. _sec-local-langlands:

.. _local-langlands:

The Local Langlands Correspondence
----------------------------------

The Local Langlands Correspondence is a canonical bijection
:math:`\pi\mapsto\sigma(\pi)` between irreducible admissible representations of
:math:`G` and local 2-dimensional representations of the absolute Galois group
of :math:`{\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 :cite:`Tate`.) The
bijection manifests as an agreement of :math:`L`- and
:math:`\varepsilon`-factors that one constructs for each category. The
foundation for the Local Langlands Correspondence for
:math:`{\operatorname{GL}}_2` over a non-archimedean field was laid
in :cite:`JacquetLanglands`; the work was completed
in :cite:`Kutzko1` and :cite:`Kutzko2`.

We remark that the conductor of :math:`\pi` agrees with the Artin conductor of
:math:`\sigma(\pi)`, and that :math:`\pi` is principal series (resp., Steinberg,
supercuspidal) if and only if :math:`\sigma(\pi)` is a sum of two characters
(resp., reducible but not decomposable, irreducible). We also remark that if
:math:`p\neq 2` and :math:`\sigma` is an irreducible 2-dimensional Galois
representation of :math:`{\mathbb{Q}}_p`, then :math:`\sigma` must be induced
from a character :math:`\chi` of a quadratic field extension
:math:`E/{\mathbb{Q}}_p`. Then :math:`(E,\chi)` is called an *admissible pair*
(see :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 :math:`\Pi` of
the adele group :math:`{\operatorname{GL}}_2({\mathbb{A}}_{\mathbb{Q}})` which
appear in the Hilbert space of square-integrable cuspidal functions
:math:`L^2_0({\operatorname{GL}}_2({\mathbb{Q}})\backslash{\operatorname{GL}}_2({\mathbb{A}}_{\mathbb{Q}}),\varepsilon)`,
where :math:`\varepsilon` is a Dirichlet character. Let :math:`f` be a cuspidal
newform for :math:`\Gamma_0(N)` with Dirichlet character :math:`\varepsilon`.
Then there is associated to :math:`f` a cuspidal automorphic representation
:math:`\Pi_f`, see :cite:`Gelbart`. This is a restricted tensor
product :math:`\bigotimes_{p\leq\infty} \pi_{f,p}`, where if :math:`p` is a
finite prime, :math:`\pi_{f,p}` is an admissible representation of
:math:`{\operatorname{GL}}_2({\mathbb{Q}}_p)`. There is also the Galois
representation :math:`\rho_f` attached to :math:`f` constructed by Deligne.
By :cite:`Carayol` there is a straightforward relationship between
:math:`\sigma(\pi_{f,p})` and the restriction of :math:`\rho_f` to the
decomposition group at :math:`p`. Therefore to determine the local properties of
:math:`\rho_f` it is enough to compute the local components :math:`\pi_{f,p}`.
These are almost always unramified principal series; the only challenge is to
compute :math:`\pi_{f,p}` when :math:`p` divides :math:`N`.

Category
--------

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

Verbose Output
--------------

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