**J. Tilouine**

Universite Paris, France

July 9, 2015

**Big image of Galois for positive slope families**:
In the 90's, Coleman and Mazur defined the Eigencurve
$X$ which parametrizes the Hecke eigensystems of modular forms
with non zero eigenvalue for $U_p$. Recently, Hida showed that non CM
ordinary families have big Galois image. We do the same, in the non
ordinary case, for families of finite slope on $X$. The ingredient
replacing ordinarity is Sen's theory in the relative setting.