Minimally ramified deformations and rigidity of conjugate self-dual automorphic Galois representations

par M. Yichao TIAN

jeudi 25 juin 2020, 14:00 → 15:00 Europe/Paris

Consider a CM extension E/F of a totally real number field, and a conjugate self-dual regular algebraic cuspidal automorphic representation Pi of GL_n(A_E). We know that, for each prime l, there is a conjugate self-dual mod l Galois representation of E attached to Pi.  Let S be a finite set of nonarchimedean places of E containing all ramified places for  Pi and E/F. In this talk, I will explain that, if Pi is supercuspidal at one nonarchimedean place,  the mod l Galois representation attached to Pi is rigid for S for almost all primes l, i.e., its restriction of the mod l representation to the local Galois group of any place v in S admits only minimally ramified deformations. As an application, one can get a R=T theorem for l-adic cohomology of unitary Shimura varieties for almost all primes l without too much restrictions. This talk is based on Appendix E of my joint work with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhu on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.