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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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