## STNB2020(34th edition)

### On local constancy phenomena for reductions of 2-dimensional crystalline representations

Emiliano Torti

#### Abstract

Irreducible crystalline representations of dimension 2 depend, up to twists, on two parameters: a weight k and a trace of a Frobenius map $a_p$. In this talk, we will study when it is true that if the parameters are sufficiently $p$-adically close, then also the correspondent representations are $p$-adically close (in the sense of isomorphic reductions). These phenomena have been extensively studied by L. Berger in the residual case. If time permits, we will see applications of such results in the context of modular forms.