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 (the restriction to inertia of) that Galois representation.
The algorithms implemented here are described in [Loeffler and Weinstein, n.d.].
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]. For an accessible introduction, see [Bushnell and Henniart, 2006].
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:
every vector \(v\in V\) is fixed by a compact open subgroup of \(G\), and
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]. 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]. 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], 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].) 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]; the work was completed in [Kutzko, 1980] and [Kutzko, 1984].
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], 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]. 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] 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).