First steps towards p-adic Langlands functoriality

By C. Breuil, P. Schneider. 2006

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Abstract By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0d1a5827ad6c1324480dfd9d7ee04047cbc8dc76

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Citations in the corpus

The local Langlands correspondence for GL_n in families (2011, unknown): “s a continuous n-dimensional representation of GE over an extension K of Qp. The correspondence ρ→ π(ρ) is essentially the generic local Langlands correspondence introduced by Breuil and Schneider in [2]. Unlike more standard formulations of local Langlands, the representation π(ρ) of GLn(E) will in”
The p-adic local Langlands correspondence for GL_2(Q_p) (2013, background): “ntation of G(so there exist a∈Zand k≥1such that W= Symk−1L2⊗deta). Let ∆ be the Weil representation corresponding to πvia the classical local Langlands correspondence; we view ∆ as a (ϕ,GQp)-module 3 [40, 13]. Let F(∆,W) be the space of isomorphism classes of weakly admissible, absolutely irreducibl”
Patching and the p-adic local Langlands correspondence (2013, methodology): “call the conjecture in more detail in Section 5 below, to which the reader should refer for any unfamiliar notation or terminology. Given a de Rham representation r: GF →GLn(Qp) of regular weight, in [BS07] there is associated to r a locally algebraic Qp-representation BS(r) of GLn(F). The following”