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 p-primary Selmer group (i.e., its $$\lambda $$ -invariant) over cyclotomic -extension is large. More precisely, any natural number n, one constructs representation such that associated -invariant $$\ge n$$ . The method based on study congruences between forms, and leverages results Greenberg Vatsal. Given form $$f_1$$ satisfying suitable conditions, congruent $$f_2$$ A key ingredient in acheiving this theoretic lifting result Fakhruddin–Khare–Patrikis, extends previous work Ramakrishna. are illustrated by explicit examples.