Volume 56, Issue 5 p. 1624-1642
RESEARCH ARTICLE

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

Anwesh Ray

Corresponding Author

Anwesh Ray

Chennai Mathematical Institute, Kelambakkam, Tamil Nadu, India

Correspondence

Anwesh Ray, Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam, Siruseri, Tamil Nadu 603103, India.

Email: [email protected]

Search for more papers by this author
First published: 26 February 2024

Abstract

Let p 5 $p\geqslant 5$ be a prime number. We consider the Iwasawa λ $\lambda$ -invariants associated to modular Bloch–Kato Selmer groups, considered over the cyclotomic Z p $\mathbb {Z}_p$ -extension of Q $\mathbb {Q}$ . Let g $g$ be a p $p$ -ordinary cuspidal newform of weight 2 and trivial nebentype. We assume that the μ $\mu$ -invariant of g $g$ vanishes, and that the image of the residual representation associated to g $g$ is suitably large. We show that for any number n $n$ greater than or equal to the λ $\lambda$ -invariant of g $g$ , there are infinitely many newforms f $f$ that are p $p$ -congruent to g $g$ , with λ $\lambda$ -invariant equal to n $n$ . We also prove quantitative results regarding the levels of such modular forms with prescribed λ $\lambda$ -invariant.