Potential automorphy and change of weight

By Thomas Barnet-Lamb, T. Gee, David Geraghty, Richard Taylor. 2010

319 citations recorded by the source corpus.

We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call \potentential diagonalizability." This result allows for \change of weight" and seems to be substantially more exible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real eld is potentially automorphic and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satises the expected functional equation.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7ba40cdc270334a6f2472f910c76377cad337fc5

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Citations in the corpus

Lattices in the cohomology of Shimura curves (2013, methodology): “1.1]. (Note that we will use the symbol ıin other contexts in the body of the paper, but we will not have to mention the isomorphism Qp→Cagain, so we have denoted it by ıhere for ease of reference to [BLGGT14].) 1.9. The inertial local Langlands correspondence. Let Lbe a local ﬁeld of characteristic”
The Sato-Tate conjecture for Hilbert modular forms (2009, unknown): “We remark that in the forthcoming paper [BLGGT10] we remove the need to consider lattices, and generalise the arguments of this paper to prove potential automorphy theorems for a broad class of Galois representations.”
Even Galois representations and the Fontaine–Mazur conjecture (2009, methodology): “The scheme Spec(R ,τ,v ρ [1/p]) is not in general smooth, so we must impose strong connectedness in the potentially semi-stable case in order for the arguments of [BLGGT10] to apply.”
On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne (2011, background): “e). Lemma 3.8. Say that a lift (ρ,R) is of type DC v if the induced map R →Rfactors through Rλ,C v . Then D C v is a local deformation problem. Proof. The proof is identical to that of Lemma 1.2.1 of [BLGGT]. Deﬁnition 3.9. Let λ ∈(Zn +) Hom(Lv,K). We say that a continuous representation ρ: GLv →GLn”
A geometric perspective on the Breuil–Mézard conjecture (2011, methodology): “We will freely use the notation and terminology of [BLGGT10].”
Serre weights for rank two unitary groups (2011, methodology): “Firstly, we have developed a considerable body of material on automorphy lifting theorems for these groups in our recent work ([BLGG11], [BLGG12], [BLGGT10]).”
On the rigid cohomology of certain Shimura varieties (2014, background): “In the presence of such a self-duality assumption (‘polarizability’, see [9]) the existence of rp,ı(π ) has been known for some years (see [18,51]).”
The buzzard-diamond-jarvis conjecture for unitary groups (2012, background): “theorems, it is not possible to directly prove that all of these weights are modular; rather, the best one can do is prove that the weights which admit potentially diagonalizable crystalline lifts (in the sense of [BLGGT10]) are modular, which is proved in [BLGG11].”
On the Sato-Tate conjecture for non-generic abelian surfaces (2013, methodology): “ are recorded in §3 of [Rib] when F= Q, but they extend with the same proofs to general totally real F. We refer to the introduction of [PT] for the deﬁnition of a weakly compatible system, see §5 of [BLGGT] for a more extensive treatment. We remark that in §3 of [Rib] it is assumed throughout that ”