Galois representations at the boundary of the eigencurve

Abstract

Modular forms are interesting objects for number theorists for many reasons. In particular, Deligne showed that Galois representations can be attached naturally to modular forms. One of the key properties of these representations, as discovered by Kisin, and Colmez later, is that these representations are `trianguline'. In particular, Kedlaya-Pottharst-Xioa and Liu independently showed that this property extends to families of Galois representations, parametrized by a geometric object that is the `eigencurve' defined by Coleman and Mazur. Andreatta, Iovita, and Pilloni have proven the existence of an `integral' eigencurve, which includes characteristic $p$ points at the boundary and contains the classical eigencurve in the interior. In joint work with Ruochuan Liu, using the theory of Crystalline periods, we show that the Galois representations associated to these characteristic $p$ points also satisfy the trianguline property.