Automorphy for some l-adic lifts of automorphic mod l Galois representations

By L. Clozel, Michael Harris, Richard Taylor. 2008

486 citations recorded by the source corpus.

We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL2. Following Wiles’ method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d0368935d7d9cb9ee6812a894db443ca3a56a2cc

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In Geometry and Topology

1. T. Gee
2. M. Emerton
3. David Geraghty
4. Thomas Barnet-Lamb
5. M. Kisin
6. F. Herzig
7. J. Thorne
8. Frank Calegari

In Mathematical Physics

1. T. Gee
2. M. Emerton
3. David Geraghty
4. M. Kisin
5. Thomas Barnet-Lamb
6. F. Herzig
7. Frank Calegari
8. J. Thorne

Citations in the corpus

• The sign of Galois representations attached to automorphic forms for unitary groups (2008, background): “More recently, this question has been extensively discussed in [ ClHT ] where some cases of the above theorem, concerning Galois representations with some constraining properties ensuring they have a nice and workable deformation theory, are proved by a very indirect method ‐ indeed the whole long a”
• Weight cycling and Serre-type conjectures for unitary groups (2011, methodology): “re induced by ı◦eτ : F →C. As usual, σλ does not depend on the choice of the eτ up to isomorphism. Let Adenote the space of automorphic forms on G(F+)\G(A F+). As in the proof of Proposition 3.3.2 of [CHT08] one easily obtains the following lemma. Lemma 7.1.6. There is an isomorphism of G(A∞ F+)-mod”
• On the infinite fern of Galois representations of unitary type (2009, methodology): “to Lemma 1.5 (using the Gn here) and is checked in [ CHT , §2].”
• THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS (2012, methodology): “ each place v∈ S p, and each homomorphism x: TT λ,τ(U,O) m → Q , x◦ρ m|GF ve is potentially crystalline of Hodge type λv and Galois type τv. Proof. This maybe provedin the samewayas Proposition3.4.4of[CHT08], making use of Corollaire 5.3 of [Lab09], Theorem 1.1 of [BLGGT12], and Theorem 1.1.3 above.”