Constructing Galois representations with large Iwasawa $\lambda$-Invariant

TL;DR

This paper constructs modular Galois representations with arbitrarily large Iwasawa $\lambda$-invariants by studying congruences between modular forms and applying advanced Galois lifting techniques.

Contribution

It introduces a method to produce modular Galois representations with large $\lambda$-invariants using congruences and Galois lifting results, extending prior work.

Findings

Constructed representations with $\lambda$-invariant $\geq n$ for any $n$

Utilized congruences between modular forms and Galois lifting techniques

Provided explicit examples illustrating the results

Abstract

Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $\lambda$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for any natural number $n$, one constructs a modular Galois representation such that the associated $\lambda$-invariant is $\geq n$. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form $f_1$ satisfying suitable conditions, one constructs a congruent modular form $f_2$ for which the $\lambda$-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples.