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

Clozel, Laurent; Harris, Michael; Taylor, Richard

Publications Mathématiques de l'IHÉS, Tome 108 (2008), pp. 1-181.

We extend the methods of Wiles and of Taylor and Wiles from ${\mathrm{GL}}_2$ 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 ${\mathrm{GL}}_2$. 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.

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