Constructing Galois representations with large Iwasawa ?-invariant

Let p >= 5 be a prime. We construct modular Galois representations for which the Z(p)-corank of the p-primary Selmer group (i.e., lambda-invariant) over the cyclotomic 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 >= 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.