**Denis Benois**

University of Bordeaux, France

February 25, 2016

**p-adic L-functions and p-adic Hodge theory**:
L-functions associated to modular (or, more generally, automorphic)
forms contain a fashinating number-theoretic information. Thanks to the work of many mathematicians we have actually a satisfactory
conjectural picture relating the so-called special values of L-functions to some number-theoretic invariants. The only known general approach to these conjectures is to construct p-adic analogs of complex L-functions which are related to the arithmetic more directly than their complex counterparts. This is the domain of Iwasawa theory.
In the first part of this talk we give a survey of know
results in this direction. In the second part, we discuss the so called trivial zero conjecture which describes the leading term of a p-adic L-function in the case of additional zeros.