On certain special values of $L$-functions associated to elliptic curves and real quadratic fields

Prof. Chung Pang Mok
2020-09-24 16:30-17:30


Title: Connectedness of Kisin varieties associated to  absolutely irreducible Galois representations

Speaker: Prof. Miaofen Chen (East China Normal University)

Time: 16:00-17:00, August 27, 2020 (Beijing Time)

Abstract: Let K be a p-adic field. Let \rho be a n-dimensional continuous absolutely irreducible mod p representation of the absolute Galois group of K. The Kisin variety is a projective scheme which parametrizes the finite flat group schemes over the ring of integers of K with generic fiber \rho satisfying some determinant condition. The connected components of the Kisin variety is in bijection with the connected components of the generic fiber of the flat deformation ring of \rho with given Hodge-Tate weights.  Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if K  is totally ramfied with n=3 or the determinant condition is of a very particular form.  We also give counterexamples to show Kisin's conjecture does not hold in general. This is a joint work with Sian Nie.