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Automorphic lifts of prescribed types. (English) Zbl 1276.11085
Let \(p > 2\) be a prime, \(F\) a totally real field, \(S\) a finite set of places of \(F\) containing the places of \(F\) dividing \(p\) and the infinite places, and \(\Sigma \subseteq S\) be the set of finite places in \(S\). Let \(E/\mathbb{Q}_p\) be a finite extension with ring of integers \(\mathcal{O}\) and residue field \(\mathbb{F}\) and \(\overline{\rho} : G_{F,S} \rightarrow \text{GL}_2(\mathbb{F})\) be a continuous residual representation.
Assume \(\overline{\rho} \mid G_{F(\zeta_p)}\) is absolutely irreducible and additionally that \([F(\zeta_p):F] \not= 2\) if \(p = 5\) and the projective image of \(\overline{\rho}\) is \(\text{PGL}_2({\mathbb F}_5)\).
Assume \(\overline{\rho} \cong \overline{\rho}_f\) is modular for \(f\) a Hilbert modular form of parallel weight \(2\), with \(f\) potentially ordinary at all places \(v \in Z\), where \(Z\) is the set of places \(v \mid p\) such that the assigned local deformation ring component at \(v\) is ordinary, and \(\pi_f\) not special at any place dividing \(p\). Assuming there are no local obstructions, it is shown that there exists a modular lifting of \(\overline{\rho}\) which is potentially Barsotti-Tate at all places \(v \mid p\), unramified outside \(S\), and which has type \(\tau_v\) at all places \(v \in \Sigma\). A consequence of this result is level lowering for places not dividing \(p > 2\).
The author describes a generalization of the Serre weight conjecture for two dimensional residual representations of \(G_F\) in [K. Buzzard, F. Diamond and F. Jarvis, Duke Math. J. 155, No. 1, 105–161 (2010; Zbl 1227.11070)] so that it includes the case when \(p\) is no longer unramified in \(F\). The author then proves the conjecture in the case when \(p > 2\) and \(p\) is completely split in \(F\).
The author states a general conjecture about the possible weights of an \(n\)-dimensional residual representation of \(G_{\mathbb Q}\). Some level-raising and level-lowering results away from \(p\) for \(n\)-dimensional residual representations of \(G_{\mathbb Q}\) are also described.

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
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