Volume 56, Issue 5 p. 1624-1642

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

Anwesh Ray

Corresponding Author

Anwesh Ray

Chennai Mathematical Institute, Kelambakkam, Tamil Nadu, India


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

Email: [email protected]

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First published: 26 February 2024


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.