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A correspondence between representations of local Galois groups and Lie-type groups. (English) Zbl 1230.11069
Burns, David (ed.) et al., \(L\)-functions and Galois representations. Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press (ISBN 978-0-521-69415-5/pbk). London Mathematical Society Lecture Note Series 320, 187-206 (2007).
Introduction: J.-P. Serre conjectured in [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] that every continuous, irreducible odd representation \(\rho: G_{\mathbb Q}\to \text{GL}_2(\mathbb F_p)\) arises from a modular form. Moreover he refines the conjecture by specifying an optimal weight and level for a Hecke eigenform giving rise to \(\rho\). Viewing Serre’s conjecture as a manifestation of Langlands’ philosophy in characteristic \(p\), this refinement can be viewed as a local-global compatibility principle, the weight of the form reflecting the behavior of \(\rho\) at \(p\), the level reflecting the behavior at primes other than \(p\). The equivalence between the “weak” conjecture and its refinement (for \(\ell >2\)) follows from work of K. A. Ribet [Invent. Math. 100, No. 2, 431–476 (1990; Zbl 0773.11039)] and others. Remarkable progress has recently been made on the conjecture itself by Khare and Wintenberger [C. Khare, J.-P. Wintenberger and M. Kisin, Serre’s modularity conjecture. I, II. Invent. Math. 178, No. 3, 485–504, 505–586 (2009; Zbl 1304.11041, Zbl 1304.11042)]; see for example C. Khare’s article in this volume [L-functions and Galois representations. London Mathematical Society Lecture Note Series 320, 270–299 (2007; Zbl 1171.11034)].
Serre’s conjecture is generalized in [K. Buzzard, F. Diamond and A. F. Jarvis, Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)] to the context of Hilbert modular forms and two-dimensional representations of \(G_K\) where \(K\) is a totally real number field in which \(p\) is unramified. The difficulty in formulating the refinement lies in the specification of the weight. This is handled in [Buzzard et al., loc. cit.] by giving a recipe for a set \(W_{\mathfrak p}(\rho)\) of irreducible \(\mathbb F_p\)-representations of \(\text{GL}_2(\mathcal O_K/\mathfrak p)\) for each prime \(\mathfrak p\mid p\) in terms of \(\rho|_{I_{\mathfrak p}}\); the sets \(W_{\mathfrak p}(\rho)\) then conjecturally characterize the types of local factors at primes over \(p\) of automorphic representations giving rise to \(\rho\). We omit the subscript \(\mathfrak p\) since we are concerned only with local behavior, so now \(K\) will denote a finite unramified extension of \(\mathbb Q_p\) with residue field \(k\).
The purpose of the paper is to prove that if the local Galois representation is semisimple, then \(W(\rho)\) is essentially the set of Jordan-Hölder constituents of the reduction of an irreducible characteristic zero representation of \(\text{GL}_2(k)\). Moreover, denoting this representation \(\alpha(\rho)\) we obtain
Theorem 1. There is a bijection
\[ \{\rho: G_K\to \text{GL}_2(\mathbb F_p)/\sim\quad\text{of } \rho|^{\text{ss}}_{I_K} \]
\[ \alpha \updownarrow \]
\[ \{\text{irreducible \(\overline{\mathbb Q}_p\)-representations of \(\text{GL}(k)\) not factoring through } \det\}/ \sim \]
\[ \text{such that \(W(\rho^{\text{ss}})\) contains the set of Jordan-Hölder factors of the reduction of } \alpha(\rho). \]
Moreover, the last inclusion is typically an equality and one can explicitly describe the exceptional weights. We remark that the local Langlands correspondence also gives rise to a bijection between the sets in the theorem by taking the \(K\)-type corresponding to a tamely ramified lift of \(\rho\). The bijection of the theorem however has a different flavor. Indeed if \([k:\mathbb F_p]\) is odd, then irreducible \(\rho\) correspond to principal series and special representations, while reducible \(\rho\) correspond to supercuspidal ones.
A generalization of Serre’s Conjecture to the setting of \(\text{GL}_n\) was formulated by Ash and others, and F. Herzig’s thesis [The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations, Ph.D. Thesis, Harvard University (2006)] pursues the idea of relating the set of Serre weights of a semi-simple \(\rho: G_{\mathbb Q_p}\to \text{GL}_2n(\overline{\mathbb F}_p)\) to the reduction of an irreducible characteristic zero representation of \(\text{GL}_n(\mathbb F_p)\). However, Herzig shows that the phenomenon described in Theorem 0.1 does not persist for \(n > 2\); instead he defines an operator \(\mathcal R\) on the irreducible mod \(p\) representations of \(\text{GL}_n(\mathbb F_p)\) and shows that the regular (i.e., up to certain exceptions) Serre weights of \(\rho\) are given by applying \(\mathcal R\) to the constituents of the reduction of a certain characteristic zero representation \(V(\rho)\). Herzig also show that such a relationship holds in the context of \(\text{GL}_2(k)\). Moreover, the association \(\rho\mapsto V(\rho)\) appears to be compatible with the local Langlands correspondence in the sense described above. In this light, Theorem 1 can be viewed as saying that Herzig’s operator \(\mathcal R\) typically sends the set of irreducible constituents of the reduction of one \(\overline{\mathbb Q}_p\)-representation of \(\text{GL}_2(k)\) to those of another.
One can also view Theorem 1 in the context of the theory of mod \(p\) and \(p\)-adic local Langlands correspondences being developed by Breuil and others. In particular, one would like a mod \(p\) local Langlands correspondence to associate a mod \(p\) representation of \(\text{GL}_2(k)\) to \(\rho\), and local-global compatibility considerations suggest that the set of Serre weights comprise the constituents of its \(\text{GL}_2(\mathcal O_K)\)-socle. One would also like a \(p\)-adic local Langlands correspondence associating \(p\)-adic representations of \(\text{GL}_2(K)\) to suitable lifts of \(\rho\), and satisfying some compatibility with the mod \(p\) correspondence with respect to reduction. One can thus speculate that the theorem reflects some property of the hypothetical \(p\)-adic correspondence for \(\text{GL}_2\).
The paper is organized as follows: In Section 1, we compute the semisimplification of the reduction mod \(p\) of the irreducible characteristic zero representations of \(\text{GL}_2(k)\). The main theorem is proved in Section 2, and the exceptional weights are described in Section 3 for the sake of completeness.
For the entire collection see [Zbl 1130.11004].

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F85 \(p\)-adic theory, local fields