We prove an exact control theorem, in the sense of Hida theory, for the ordinary part of the middle degree étale cohomology of certain Hilbert modular varieties, after localizing at a suitable maximal ideal of the Hecke algebra. Our method of proof builds upon the techniques introduced by Loeffler–Rockwood–Zerbes (2023, Spherical varieties and p-adic families of cohomology classes); another important ingredient in our proof is the recent work of Caraiani–Tamiozzo (2023, Compositio Mathematica 159, 2279–2325) on the vanishing of the étale cohomology of Hilbert modular varieties with torsion coefficients outside the middle degree. This work will be used in forthcoming work of the author to show that the Asai–Flach Euler system corresponding to a quadratic Hilbert modular form varies in Hida families.

We study the variation of $\unicode[STIX]{x1D707}$-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$-adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$-adic $L$-function is simply a power of $p$ up to a unit (i.e. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

This paper is a continuation of the author's previous work, where we studied one of the inequalities between the characteristic ideal of the Selmer group and the ideal of the $p$-adic $L$-function predicted by the two-variable Iwasawa main conjecture for a nearly ordinary Hida deformation $\mathcal{T}$. In this paper, we study several properties of the Selmer group and the $p$-adic $L$-function solving some of the open questions raised in the author's previous work. As applications, we have an infinite family of elliptic cuspforms where the cyclotomic Iwasawa main conjecture holds for every cuspform in the family.

We build a theory of Λ-adic Siegel modular forms related to the Klingen parabolic subgroup of GSp(4). These correspond to families of cohomology classes of increasing levels whose Hecke eigenvalues enjoy strong congruence properties. In the spirit of Hida‘s theory, a control theorem to relate the family to finite-level members is proved for almost all primes p; in particular we show that the error term appearing in degree one cohomology is killed by the ordinary idempotent.