On the finiteness of patched completed cohomology
Shen-Ning Tung
2019-07-02 15:30-16:30
MCM110
Title: Breuil-Kisin modules and integral p-adic Hodge theory
Speaker: Hui Gao (University of Helsinki)
Time: 2019-07-02, 14:00-15:00
Place: MCM110
Abstract: We construct a certain variant of Liu's $(\varphi, \hat{G})$-modules to classify integral semi-stable Galois representations. Our theory uses Breuil-Kisin modules and Breuil-Kisin-Fargues modules with Galois actions, and can be regarded as the algebraic avatar of the integral p-adic cohomology theories of Bhatt-Morrow-Scholze and Bhatt-Scholze. Along the way, we classify Galois representations that are finite height in terms of Breuil-Kisin modules.
Title: On the finiteness of patched completed cohomology
Speaker: Shen-Ning Tung (University Duisburg-Essen)
Time: 2019-07-02, 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.
