Big image of Galois representations associated with finite slope $p$-adic families of modular forms

Abstract

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.Comment: 23 pages; revision of Section 2 (see Remark 2.4) and improvement of Proposition 4.14, plus minor changes. Published in "Elliptic Curves, Modular Forms and Iwasawa Theory. In Honour of John H. Coates' 70th Birthday, Cambridge, UK, March 2015", Springer Proceedings in Mathematics & Statistics, Vol. 188, 201