an:06185838
Zbl 1276.11102
Harris, Michael
The Taylor-Wiles method for coherent cohomology
EN
J. Reine Angew. Math. 679, 125-153 (2013).
00320628
2013
j
11G18 11F80
Galois representations; Taylor-Wiles method; Shimura varieties; automorphic vector bundles
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 \textit{K.-W. Lan} and \textit{J. Suh} [Int. Math. Res. Not. 2011, No. 8, 1870--1879 (2011; Zbl 1233.11042)] as well as [\textit{K.-W. Lan} and \textit{J. Suh}, ``Vanishing theorems for torsion automorphic sheaves on compact PEL-type Shimura varieties'', manuscript (2010)], which provide a vanishing theorem ?? la [\textit{H. Esnault} and \textit{E. Viehweg}, Lectures on vanishing theorems. DMV Seminar. 20. Basel: Birkh??user 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 (\S 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.
Wen-Wei Li (Beijing)
Zbl 1233.11042; Zbl 0779.14003; Zbl 1255.11027