Lecture----数学院晨兴数学中心

Title:	On the finiteness of patched completed cohomology
Speaker:	Shen-Ning Tung (University Duisburg-Essen)
Time:	2019-7-2, 15:30-16:30
Place:	MCM110
Abstract:	We prove that after applying the generalized Colmez functor constructed by Zábrádi, the patched completed cohomology with an action of \prod^r_{i=1} GL_2(Q_p) is finite over the patched Galois deformation ring. This result has the following two applications. First of all, it gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of Q_p, which is new in the case p = 2 or 3 and \overline{r} a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture by Kisin, Hu-Tan and Paskunas is removed. Secondly, it gives another proof of the 'big R = big T' theorem of Gee-Newton without the formally smoothness assumption at p.
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