Let $(R,\mathfrak{m})$ be a local ring, and let $M$ and $N$ be finite $R$-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules ${Ext}_R^i(M,N/{\mathfrak{m}}^{n}N)$. A number of corollaries are given, and more general filtrations are also considered.

A conjecture of Bonnafé, Geck, Iancu, and Lam parametrizes Kazhdan-Lusztig left cells for unequal-parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by Lusztig and the recent work of Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.

Let $X$ be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic $p$ big enough. Given a local ring ${\mathcal{O}}_{P,X}$ at a rational singular point $P$ of $X$, we attached a universal zeta function which is a rational function and admits a functional equation if ${\mathcal{O}}_{P,X}$ is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

Let $\mathcal{X}$ be a space of homogeneous type in the sense of Coifman and Weiss, and let $\mathcal{D}$ be a collection of balls in $\mathcal{X}$. The authors introduce the localized atomic Hardy space $H_{\mathcal{D}}^{p,q}\left(\mathcal{X}\right)$ with $p\in (0,1]$ and $q\in [1,\infty ]\cap (p,\infty ]$, the localized Morrey-Campanato space ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$, and the localized Morrey-Campanato-BLO (bounded lower oscillation) space ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ with $\alpha \in \mathbb{R}$ and $p\in (0,\infty )$, and they establish their basic properties, including $H_{\mathcal{D}}^{p,q}\left(\mathcal{X}\right)=H_{\mathcal{D}}^{p,\infty }\left(\mathcal{X}\right)$ and several equivalent characterizations for ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$. In particular, the authors prove that when $\alpha >0$ and $p\in [1,\infty )$, then ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)={\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)={Lip}_{\mathcal{D}}(\alpha ;\mathcal{X})$, and when $p\in (0,1]$, then the dual space of $H_{\mathcal{D}}^{p,\infty }\left(\mathcal{X}\right)$ is ${\mathcal{E}}_{\mathcal{D}}^{1/p-1,1}\left(\mathcal{X}\right)$. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$, respectively, by ${\mathcal{E}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$, when $\mathcal{D}$ is determined by $\rho$. The authors then obtain the boundedness from ${\mathcal{E}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ to ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley $g$-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ${\mathbb{R}}^d$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of $I$-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

This paper examines the moduli spaces of log Hodge structures introduced by Kato and Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. This paper treats the following two cases: (A) where the period domain is Hermitian symmetric, and (B) where the Hodge structures are of the mirror quintic type. Especially it addresses a property of the torsor.