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.

Given a reductive group $G$ and a reductive subgroup $H$, both defined over a number field $F$, we introduce the notion of the $H$-distinguished automorphic spectrum of $G$ and analyze it for the pairs $({GL}_{2n},{Sp}_n)$ and $({Sp}_{2n},{Sp}_n\times {Sp}_n)$. In the first case we give a complete description by using results of Jacquet and Rallis as well as Offen and Yamana. In the second case we give an upper bound, generalizing vanishing results of Ash, Ginzburg, and Rallis, and a lower bound, extending results of Ginzburg, Rallis, and Soudry.

Let $p\ne \ell$ be primes. We study the étale cohomology $H_{\text{ét}}^{\ast }\left({\mathrm{BGL}}_n\right({\mathbb{F}}_{p^s});\mathbb{Z}/\ell )$ by using the stratification methods from Molina-Rojas and Vistoli. To compute this cohomology, we use the Dickson algebra and the Drinfeld space.

We prove a generalization of the Jones–Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several examples that do not satisfy a naive generalization of the Jones–Kawamuro conjecture.