The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplication

Abstract

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