Congruences between Hilbert modular forms of weight $2$, and the Iwasawa $\lambda$-invariants

TL;DR

This paper proves the equality of algebraic and analytic Iwasawa lambda-invariants for certain Hilbert modular forms, extending previous results to cases with reducible residual Galois representations.

Contribution

It generalizes Greenberg and Vatsal's result to Hilbert cusp forms of weight 2 with reducible residual Galois representations at an ordinary prime.

Findings

Equality of algebraic and analytic lambda-invariants established

Extends previous results to reducible residual Galois representations

Applicable to Hilbert cusp forms of parallel weight 2

Abstract

The purpose of this paper is to prove the equality between the algebraic Iwasawa $\lambda$-invariant and the analytic Iwasawa $\lambda$-invariant for a Hilbert cusp form of parallel weight $2$ at an ordinary prime $p$ when the associated residual Galois representation is reducible. This is a generalization of a result of R. Greenberg and V. Vatsal.