-Invariants of Adjoint Square Galois Representations Coming from Modular Forms

Abstract

Let be a newform that is p-ordinary (i.e. is coprime to p), and the associated Galois representation. We find the special value ⁠. We define the analytic -invariant of a “motivic” Galois representation, and show how this special value relates to work of Greenberg and Hida on finding ⁠. In particular, we reduce finding this value to showing an equality of p-adic L-functions similar to a well-known relation of archimedean L-functions. This provides evidence for a conjecture of Greenberg, which is originally based on a conjecture of Mazur, Tate, and Teitelbaum.