Documentation

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

The algorithms implemented here are described in [LW].

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 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 ρ_ffrom Gal(/line(Q)/Q)toGL₂(/line(Q)_ell). For example, for a prime p different from ell, one may determine the restriction of ρ_f 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 Q_p. The first systematic study of admissible representations is [JL70]. For an accessible introduction, see [BH06].

Let G be the locally compact group GL₂(Q_p). 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, V^K is finite-dimensional.

The center of G is Q_p^x. If π is an irreducible admissible representation of G, then it has a unique central character εfrom Q_p^xto C^x such that π(g) acts as the scalar ε(g) for all g∈Q_p^x. The conductor of an irreducible admissible representation π is 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
c & d ) ∈GL₂(Z_p) with c ≡ 0 mod (pⁿ). If π admits a nonzero vector fixed by K₀(1)=GL₂(Z_p), then π is called spherical or unramified principal series and has conductor 1. If π admits a nonzero vector v for which π( pmatrix( a & b
c & d ) )v=ε(a)v for all pmatrix( a & b
c & d ) ∈K₀(pⁿ), and n≥1 is minimal for this condition, then the conductor of π is 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 χ is a character of Q_p^x then let π tensor χ be the representation g |-> χ(g)π(g); such a representation is a twist of χ. If π has minimal conductor among all its twists π tensor χ, then π 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 χ₁ and χ₂ be two characters of Q_p^*. Let B⊂G be the Borel subgroup of upper triangular matrices. Then pmatrix( a & b
0 & d ) |-> |a/d|^{- 1/2}χ₁(a)χ₂(d) is a character χ of B. An admissible representation π is a principal series representation if it is a composition factor of the induced representation π(χ₁, χ₂):=Ind_B^G χ. This induced representation is already irreducible unless χ₁χ₂^{- 1} equals |.|^{∓ 1}, in which case it has length two, with one 1-dimensional and one infinite-dimensional composition factor. For instance, Ind_B^G 1 has a trivial 1-dimensional submodule and an irreducible infinite-dimensional quotient 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 π(χ₁, χ₂), where χ₁ and χ₂ are unramified characters of Q_p^x (meaning they are trivial on Z_p^x).

The central character of π(χ₁, χ₂) is χ₁χ₂, and its conductor is the product of the conductors of the χ_i. Note that π(χ₁, χ₂) is minimal if and only if one of the characters χ₁, χ₂ is unramified. The Steinberg representation 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 π has conductor p^c, where c≥2. There is a convenient, if technical, classification of supercuspidal representations of G. Let π be supercuspidal. By [BH06], Ch. 15, there is a representation Ξ of an open and compact-mod-center subgroup K⊂G for which π=Ind_K^GΞ. If c is even, then we may take K to be Q_p^xGL₂(Z_p), and if c is odd, we may take K to be the normalizer of the Iwahori subgroup K₀(p) = pmatrix( Z_p^x & Z_p
pZ_p & Z_p^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 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 [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 π agrees with the Artin conductor of σ(π), and that π is 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 σ is an irreducible 2-dimensional Galois representation of Q_p, then σ must be induced from a character χ of a quadratic field extension E/Q_p. Then (E, χ) 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 Π of the adele group GL₂(A_Q) which appear in the Hilbert space of square-integrable cuspidal functions L²₀(GL₂(Q)\GL₂(A_Q), eps), where eps is a Dirichlet character. Let f be a cuspidal newform for Γ₀(N) with Dirichlet character ε. Then there is associated to f a cuspidal automorphic representation Π_f, see [Gel75]. This is a restricted tensor product bigotimes_p≤∞ π_{f, p}, where if p is a finite prime, π_{f, p} is an admissible representation of GL₂(Q_p). There is also the Galois representation ρ_f attached to f constructed by Deligne. By [Car83] there is a straightforward relationship between σ(π_{f, p}) and the restriction of ρ_f to the decomposition group at p. Therefore to determine the local properties of ρ_f it is enough to compute the local components π_{f, p}. These are almost always unramified principal series; the only challenge is to compute π_{f, p} when p divides N.

Verbose Output

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

V2.28, 13 July 2023