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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 Qp. One may compute (the restriction to inertia of) that Galois representation.

The algorithms implemented here are described in [LW].

Subsections

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 GL2 and F is the p-adic field Qp. 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 ρffrom Gal(/line(Q)/Q)toGL2(/line(Q)ell). For example, for a prime p different from ell, one may determine the restriction of ρf to the decomposition group at p.

Definitions

Here we introduce the category of admissible irreducible representations of GL2 over a non-archimedean field, which for our purposes will be Qp. The first systematic study of admissible representations is [JL70]. For an accessible introduction, see [BH06].

Let G be the locally compact group GL2(Qp). An admissible representation of G on a complex vector space V is a homomorphism πfrom G to Aut V which satisfies the properties:

(i)
every vector v∈V is fixed by a compact open subgroup of G, and
(ii)
for every compact open subgroup K⊂G, VK is finite-dimensional.

The center of G is Qp x . If πis an irreducible admissible representation of G, then it has a unique central character varepsilon from Qp x to C x such that π(g) acts as the scalar varepsilon(g) for all g∈Qp x . The conductor of an irreducible admissible representation πis a measure of how small a compact open subgroup K⊂G must be before one sees nonzero K-invariant vectors; see [Cas73]. Consider the filtration K0(pn) of subgroups of G, where K0(pn) is the subgroup of matrices pmatrix( a & b cr c & d ) ∈GL2(Zp) with c = 0 mod (pn). If πadmits a nonzero vector fixed by K0(1)=GL2(Zp), then πis called spherical or unramified principal series and has conductor 1. If πadmits a nonzero vector v for which π( pmatrix( a & b cr c & d ) )v=varepsilon(a)v for all pmatrix( a & b cr c & d ) ∈K0(pn), and n≥1 is minimal for this condition, then the conductor of πis pn. (Note the similarity with the convention used to define the level of a modular form for Γ0(N).) In both cases the vector v so described is unique up to scaling; see [Cas73]. We shall call v a new vector for π.

If chi is a character of Qp x then let πtensor chi be the representation g |-> chi(g)π(g); such a representation is a twist of chi. If πhas minimal conductor among all its twists πtensor chi, then πis 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 chi1 and chi2 be two characters of Qp^ *. Let B⊂G be the Borel subgroup of upper triangular matrices. Then pmatrix( a & b cr 0 & d ) |-> |a/d| - 1/2chi1(a)chi2(d) is a character chi of B. An admissible representation πis a principal series representation if it is a composition factor of the induced representation π(chi1, chi2):=IndBG chi. This induced representation is already irreducible unless chi1chi2 - 1 equals |.|+- 1, in which case it has length two, with one 1-dimensional and one infinite-dimensional composition factor. For instance, IndBG 1 has a trivial 1-dimensional submodule and an irreducible infinite-dimensional quotient StG, 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 π(chi1, chi2), where chi1 and chi2 are unramified characters of Qp x (meaning they are trivial on Zp x ).

The central character of π(chi1, chi2) is chi1chi2, and its conductor is the product of the conductors of the chii. Note that π(chi1, chi2) is minimal if and only if one of the characters chi1, chi2 is unramified. The Steinberg representation StG 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 πhas conductor pc, where c≥2. There is a convenient, if technical, classification of supercuspidal representations of G. Let πbe supercuspidal. By [BH06], Ch. 15, there is a representation Ξof an open and compact-mod-center subgroup K⊂G for which π=IndKGΞ. If c is even, then we may take K to be Qp x GL2(Zp), and if c is odd, we may take K to be the normalizer of the Iwahori subgroup K0(p) = pmatrix( Zp x & Zp cr pZp & Zp x ) in G. We call the pair (K, Ξ) a cuspidal inducing datum.

The Local Langlands Correspondence

The Local Langlands Correspondence is a canonical bijection π|-> σ(π) between irreducible admissible representations of G and local 2-dimensional representations of the absolute Galois group of Qp 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 [Tat79].) The bijection manifests as an agreement of L- and eps-factors that one constructs for each category. The foundation for the Local Langlands Correspondence for GL2 over a non-archimedean field was laid in [JL70]; the work was completed in [Kut80] and [Kut84].

We remark that the conductor of πagrees with the Artin conductor of σ(π), and that πis principal series (resp., Steinberg, supercuspidal) if and only if σ(π) is a sum of two characters (resp., reducible but not decomposable, irreducible). We also remark that if p != 2 and σis an irreducible 2-dimensional Galois representation of Qp, then σmust be induced from a character chi of a quadratic field extension E/Qp. Then (E, chi) is called an admissible pair (see [BH06], 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 Πof the adele group GL2(AQ) which appear in the Hilbert space of square-integrable cuspidal functions L20(GL2(Q)\GL2(AQ), eps), where eps is a Dirichlet character. Let f be a cuspidal newform for Γ0(N) with Dirichlet character varepsilon. Then there is associated to f a cuspidal automorphic representation Πf, see [Gel75]. This is a restricted tensor product bigotimesp≤∞ πf, p, where if p is a finite prime, πf, p is an admissible representation of GL2(Qp). There is also the Galois representation ρf attached to f constructed by Deligne. By [Car83] there is a straightforward relationship between σ(πf, p) and the restriction of ρf to the decomposition group at p. Therefore to determine the local properties of ρf it is enough to compute the local components πf, p. These are almost always unramified principal series; the only challenge is to compute πf, 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).

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013