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The Taylor-Wiles method for coherent cohomology. (English) Zbl 1276.11102
The upshot of the Taylor-Wiles method in their proof of Fermat’s Last Theorem is the so-called $$R=T$$ theorem, where $$R$$ is the deformation ring of mod $$p$$ Galois representations and $$T$$ is a ring of Hecke operators. This technique has been improved independently by Diamond and Fujiwara since then. It is based on a comparison of modules of automorphic cohomology over $$p$$-adic integers.
In this paper, the author shows that the Diamond-Fujiwara method can also be applied by replacing the topological cohomology by coherent cohomology of suitable automorphic vector bundles. One of the main ingredients is the works of K.-W. Lan and J. Suh [Int. Math. Res. Not. 2011, No. 8, 1870–1879 (2011; Zbl 1233.11042)] as well as [K.-W. Lan and J. Suh, “Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties”, manuscript (2010)], which provide a vanishing theorem à la [H. Esnault and E. Viehweg, Lectures on vanishing theorems. DMV Seminar. 20. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)] for automorphic vector bundles on Shimura varieties of PEL-type, under certain regularity and $$p$$-smallness conditions.
On the other hand, to start the Diamond-Fujiwara machine, one also need to verify the Galois hypotheses (§4.3). Results for unitary groups obtained by the French school are used; an excellent reference thereof is the Book Project [Stabilization of the trace formula, Shimura varieties, and arithmetic applications. Volume 1: On the stabilization of the trace formula. Somerville, MA: International Press (2011; Zbl 1255.11027)].
As the author pointed out, although there is no new result about Galois representations, the bonus is that in the course of proving $$R=T$$, one obtains the freeness of $$H^{q(\mathcal{F})}(\mathbb{S}_K, \mathcal{F})$$ over the localized Hecke algebra. Some remarks about (i) the case of non-compact Shimura varieties (for which one might need the “interior cohomology”) and (ii) about the extension to Hida families are also given.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11F80 Galois representations
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