The Taylor-Wiles method for coherent cohomology.

*(English)*Zbl 1276.11102The 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.

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.

Reviewer: Wen-Wei Li (Beijing)