Constructing Galois representations with prescribed Iwasawa λ$\lambda$‐invariant

Abstract

AbstractLet be a prime number. We consider the Iwasawa ‐invariants associated to modular Bloch–Kato Selmer groups, considered over the cyclotomic ‐extension of . Let be a ‐ordinary cuspidal newform of weight 2 and trivial nebentype. We assume that the ‐invariant of vanishes, and that the image of the residual representation associated to is suitably large. We show that for any number greater than or equal to the ‐invariant of , there are infinitely many newforms that are ‐congruent to , with ‐invariant equal to . We also prove quantitative results regarding the levels of such modular forms with prescribed ‐invariant.