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The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations. (English) Zbl 1232.11065

J.-P. Serre conjectured in 1973 that every two-dimensional irreducible, odd Galois representation arises from a modular eigenform, and predicted later in [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] the level and weight for some such an eigenform. Serre’s conjecture has now been proved by C. Khare and J.-P. Wintenberger [“Serre’s modularity conjecture. I, II”, Invent. Math. 178, No. 3, 485–504 (2009; Zbl 1304.11041), ibid. 505–586 (2009; Zbl 1304.11042)] and M. Kisin [“Modularity of 2-adic Barsotti-Tate representations”, Invent. Math. 178, No. 3, 587–634 (2009; Zbl 1304.11043)].
Consider now \(n\)-dimensional irreducible, odd Galois representations \(\rho: \text{Gal}(\overline {\mathbb Q}/{\mathbb Q}) \to GL_n(\overline{{\mathbb F}}_p)\). A. Ash, D. Doud and D. Pollack [Duke Math. J. 112, No.3, 521–579 (2002; Zbl 1023.11025)] and A. Ash and W. Sinnott [Duke Math. J. 105, No. 1, 1–24 (2000; Zbl 1015.11018)] conjectured that such \(\rho\) arise in the mod \(p\) cohomology of \(\Gamma_1(N^{?}(\rho)) \leq \text{SL}_n({\mathbb Z})\), where \(N^{?}(\rho)\) is the Artin conductor of \(\rho\). The eigenvectors in mod \(p\) cohomology under a natural Hecke action are group cohomology with coefficients in irreducible modules of \(\text{GL}_n(\overline{{\mathbb F}}_p\) over \(\overline{{\mathbb F}}_p\), on which \(\Gamma_1(N^{?}(\rho))\) acts via reduction modulo \(p\)), and which are called Serre weights. The author’s Conjecture 1.1 predicts the set of all regular Serre weights that can occur for an \(n\)-dimensional irreducible, odd, and tamely ramified at \(p\), Galois representation \(\rho\). When \(n = 3\), this predicted set of regular Serre weights contains the weights predicted by Ash, Doud, Pollack, and Sinnott as a proper subset, and computational evidence for the extra weights is provided by calculations of Doud and Pollack. The author also obtains theoretical evidence for the conjecture in the case \(n = 4\). He also discusses the compatibility of his conjecture with conjectures of T. Gee [Math. Ann. 350, No. 1, 107–144 (2011; Zbl 1276.11085)] and of K. Buzzard, F. Diamond and F. Jarvis [Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)].

MSC:

11F80 Galois representations
20C33 Representations of finite groups of Lie type
11F75 Cohomology of arithmetic groups
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References:

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