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

Abstract

Let $p\ge 5$ 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 $\ge n$. 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.

Version 3: Final version, accepted for publication in Annales Math Quebec

References in corpus (2)

• Iwasawa Theory for Modular Forms at Supersingular Primes
• Wach modules and Iwasawa theory for modular forms

Cited by in corpus (1)

• Constructing Galois representations with prescribed Iwasawa $\lambda$-invariant