By using “Gauss sum-type” Kolyvagin systems, Kurihara studied the higher Fitting ideals of Iwasawa modules arising from Greenberg Selmer groups of p-adic Galois representations, and proved a refinement of the Iwasawa main conjecture. In this article, we study the higher Fitting ideals of Iwasawa modules arising from the dual fine Selmer groups of general Galois representations which have rank one Euler systems of “Rubin-type” circular units or Beilinson–Kato elements. By using Kolyvagin derivatives, we construct an ascending filtration ${\{{\mathfrak{C}}_i(\mathbf{c})\}}_{i\ge 0}$ of the Iwasawa algebra, and we show that the filtration ${\{{\mathfrak{C}}_i(\mathbf{c})\}}_{i\ge 0}$ gives good approximation of the higher Fitting ideals of the Iwasawa module under the assumption analogous to the Iwasawa main conjecture.

We investigate spectral properties of explosive symmetric Markov processes. Under a condition on its 1-resolvent, we prove the $L^1$-semigroups of Markov processes become compact operators.

Using a representation theoretic parameterization for the orbits in the enhanced cyclic nilpotent cone as derived by the authors in a previous article, we compute the fundamental group of these orbits. This computation has several applications to the representation theory of the category of admissible $\mathcal{D}$-modules on the space of representations of the framed cyclic quiver. First and foremost, we compute precisely when this category is semisimple. We also show that the category of admissible $\mathcal{D}$-modules has enough projectives. Finally, the support of an admissible $\mathcal{D}$-module is contained in a certain Lagrangian in the cotangent bundle of the space of representations. Thus, taking these characteristic cycles defines a map from the K-group of the category of admissible $\mathcal{D}$-modules to the $\mathbb{Z}$-span of the irreducible components of this Lagrangian. We show that this map is always injective, and is a bijection if and only if the monodromicity parameter is integral.

Given a closed subgroup $G\subset U_N^{+}$ which is homogeneous, in the sense that we have $S_N\subset G\subset U_N^{+}$, the corresponding Tannakian category C must satisfy $span({\mathcal{NC}}_2)\subset C\subset span(P)$. Based on this observation, we construct a certain integer $p\in \mathbb{N}\cup \{\mathrm{\infty }\}$, that we call the easiness level of G. The value $p=1$ corresponds to the case where G is easy, and we explore here, with some theory and examples, the case $p>1$. As a main application, we show that $S_N\subset S_N^{+}$ and other liberation inclusions, known to be maximal in the easy setting, remain maximal at the easiness level $p=2$ as well.

We prove the existence of a global family with natural universal properties over every component of the moduli space of marked irreducible holomorphic symplectic manifolds. The analogous result follows for Teichmüller spaces.