Constructing Galois representations with large Iwasawa \(\lambda \)-invariant

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., its \(\lambda \)-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 Fakhruddin–Khare–Patrikis, which extends previous work of Ramakrishna. The results are illustrated by explicit examples.