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Serre weights for quaternion algebras. (English) Zbl 1282.11042
Let \(F\) be a totally real number field with absolute Galois group \(G_F = \text{Gal}(\overline{F}/F)\). Let \(p\) be a rational prime. The authors consider the problem of giving an explicit description of the set \(W(\overline{\rho})\) of all possible weights of an irreducible mod \(p\) Galois representation \(\overline{\rho}: G_F \rightarrow \mathrm{GL}_{2}(\overline{\mathbb F}_p)\) arising from an automorphic representation on a totally definite quaternion algebra defined over \(F\) which is ramified at each place \(v \mid p\) in \(F\). Thus, the authors formulate (and for the most part prove) an analogue of Serre’s weight conjecture for automorphic forms on totally definite quaternion algebras ramified at all places above \(p\). This problem has various antecedents in the literature. For instance, it appears in the special case where \(F\) is the rational number field in [C. Khare, Invent. Math. 143, No. 1, 129-155 (2001; Zbl 0971.11028)], and is treated in the analogous setting of Hilbert modular forms with each prime above \(p\) being unramified in the work of K. Buzzard, F. Diamond and F. Jarvis [Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)]. The striking feature about the situation here, as the authors explain nicely in \(\S 3\) (see Lemma 3.3) using the Jacquet-Langlands correspondence and the local Langlands correspondence, is that the property of being modular of some specific given weight is equivalent to the representation having a lift of some specific type. This reduces the study to describing all possible types of liftings, and can be made very explicit via certain computations involving Breuil modules with descent data in the event that the restriction of \(\overline{\rho}\) to decomposition groups at primes dividing \(p\) are all semisimple. A summary of these local computations is given in \(\S 7\), leading to the final result in \(\S 8\), i.e. [Theorem 8.3]. The technical conditions required to establish that their proposed (conjectural) set of weights \(W^{?}(\overline{\rho})\) is equal to \(W(\overline{\rho})\) are essentially determined by their usage of modularity lifting theorems in the style of T. Gee [Invent. Math. 184, No. 1, 1–46 (2011; Zbl 1280.11029)] and M. Kisin [Ann. Math. (2) 170, No. 3, 1085–1180 (2009; Zbl 1201.14034)]. Thus, they must assume that \(p\) is odd and that the restriction of \(\overline{\rho}\) to the cyclotomic field \(F(\zeta_p)\) is irreducible; if \(p = 5\), they must assume additionally that the projective image of \(\overline{\rho}\) is isomorphic to \(\mathrm{PGL}_{2}({\mathbb F}_5)\) and that \([F(\zeta_p):F] =4\). Granted these conditions, along with the condition that for each prime \(v \mid p\) the restriction of \(\overline{\rho}\) to the decomposition group at \(v\) is not a twist of an extension of the trivial character by the cyclotomic character, they establish that \(W(\overline{\rho}) = W^{?}(\overline{\rho})\).

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
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