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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].

##### MSC:
 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