$\lambda$-invariant stability in families of modular Galois representations

TL;DR

This paper investigates how the Iwasawa λ-invariants vary within families of weight 2 modular forms sharing the same residual Galois representation, providing quantitative bounds on their stability and growth.

Contribution

It offers new lower bounds on the frequency of λ-invariant stability or growth in families of modular forms with identical residual Galois representations.

Findings

Lower bounds on the frequency of λ-invariant growth

Quantitative analysis of λ-invariant stability

Insights into the variation patterns of Galois representations

Abstract

Consider a family of modular forms of weight 2, all of whose residual $\pmod{p}$ Galois representations are isomorphic. It is well-known that their corresponding Iwasawa $\lambda$-invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these $\lambda$-invariants grow or remain stable.