Iwasawa theory for symmetric powers of CM modular forms at nonordinary primes, II

TL;DR

This paper advances Iwasawa theory for symmetric powers of CM modular forms at supersingular primes by proving a Main Conjecture linking p-adic L-functions and Selmer modules, extending Rubin's results to inert primes.

Contribution

It proves a Main Conjecture for symmetric powers of CM modular forms at supersingular primes, improving Rubin's results for inert primes.

Findings

Proves a Main Conjecture equating p-adic L-functions to Selmer module characteristic ideals.

Extends Rubin's results on Iwasawa theory for imaginary quadratic fields to inert primes.

Constructs finite-slope Selmer modules for symmetric powers of CM modular forms.

Abstract

Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the "admissible" $p$-adic $L$-functions to characteristic ideals of "finite-slope" Selmer modules constructed by the second author. As a key ingredient, we improve Rubin's result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.