  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.
