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

TL;DR

This paper investigates the Galois representations linked to finite slope $p$-adic families of modular forms, establishing the presence of a significant congruence Lie subalgebra within the image's Lie algebra and characterizing its maximal level.

Contribution

It proves the Lie algebra of the Galois representation's image contains a non-trivial congruence subalgebra and describes its maximal level via congruences with $p$-adic CM forms.

Findings

Lie algebra of the Galois image contains a congruence Lie subalgebra.

Largest such level is characterized by congruences with $p$-adic CM forms.

Provides structural insights into Galois representations 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.