Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes

Abstract

Our primary goal in this article is to study the Iwasawa theory for semi-ordinary families of automorphic forms on ${\mathrm{GL}}_2\times {\mathrm{Res}}_{K/\mathbb{Q}}{\mathrm{GL}}_1$, where $K$ is an imaginary quadratic field where the prime $p$ is inert. We prove divisibility results towards Iwasawa main conjectures in this context, utilizing the optimized signed factorization procedure for Perrin-Riou functionals and Beilinson--Flach elements for a family of Rankin--Selberg products of $p$-ordinary forms with a fixed $p$-non-ordinary modular form. The optimality enables an effective control on the $\mu$-invariants of Selmer groups and $p$-adic $L$-functions as the modular forms vary in families, which is crucial for our patching argument to establish one divisibility in an Iwasawa main conjecture in three variables.

A superfluous file not meant for publication has been removed from the previous version

References in corpus (3)

• Katz type p-adic L-functions for primes p non-split in the CM field
• P-adic L-functions in universal deformation families
• Rank--two Euler systems for symmetric squares