The cyclotomic Iwasawa main conjecture for Hilbert cuspforms with complex multiplication

TL;DR

This paper proves the cyclotomic Iwasawa main conjecture for Hilbert modular cuspforms with complex multiplication by deriving it from the multivariable main conjecture for CM fields, analyzing p-adic L-functions and Selmer groups.

Contribution

It establishes the main conjecture for a new class of automorphic forms by connecting it to the multivariable conjecture for CM fields, with detailed study of p-adic invariants.

Findings

Main conjecture proven for Hilbert cuspforms with CM

Analysis of p-adic L-functions under specialization

Behavior of Selmer groups in CM number fields

Abstract

We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular cuspforms with complex multiplication from the multivariable main conjecture for CM number fields. To this end, we study in detail the behaviour of the $p$-adic $L$-functions and the Selmer groups attached to CM number fields under specialisation procedures.