Hara and Yoshida introduced a notion of $\mathfrak{a}$-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from the point of view of test ideals. We also consider variants with positive real exponents.

A Brody curve is a holomorphic map from the complex plane $\mathbb{C}$ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its "mean dimension". We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension.

Let $L$ be a finite Galois extension of a number field $K$. Let $G := Gal(L/K)$. Let $z_{1}, \dots, z_{N} \in L^{*} \setminus \{1\}$ and let $m_{1}, \dots, m_{N} \in \mathbb{Q}_{l}$. Let us assume that the linear combination of $l$-adic polylogarithms $c_{n} := \sum_{i=1}^{N} m_{i}l_{n}(z_{i})_{\gamma_{i}}$ (constructed in some given way) is a cocycle on $G_{L}$ and that the formal sum $\sum_{i=1}^{N} m_{i}[z_{i}]$ is $G$-invariant. Then we show that $c_{n}$ determines a unique cocycle $s_{n}$ on $G_{K}$. We also prove a weak version of Zagier conjecture for $l$-adic dilogarithm. Finally we show that if $c_{2}$ is "motivic" ($m_{1}, \dots, m_{N} \in \mathbb{Q}$) then $s_{2}$ is also "motivic".

We show equi-distribution properties of values for the third and the fifth Painlevé transcendents in a sectorial domain. For our purpose we define a characteristic function of sectorial domain type by employing value distribution theory in a half plane. Some special cases admit analogues of Borel exceptional values. Similar results are obtained for modified versions of these Painlevé transcendents, which are of infinite growth order.

Let $R$ be a noetherian commutative ring, and

\mathbb{F} : \cdots \rightarrow F_{2} \rightarrow F_{1} \rightarrow F_{0} \rightarrow 0

a complex of flat $R$-modules. We prove that if $\kappa(\mathfrak{p}) \otimes_{R} \mathbb{F}$ is acyclic for every $\mathfrak{p} \in \operatorname{Spec} R$, then $\mathbb{F}$ is acyclic, and $H_{0}(\mathbb{F})$ is $R$-flat. It follows that if $\mathbb{F}$ is a (possibly unbounded) complex of flat $R$-modules and $\kappa(\mathfrak{p}) \otimes_{R} \mathbb{F}$ is exact for every $\mathfrak{p} \in \operatorname{Spec} R$, then $\mathbb{G} \otimes_{R}^{\bullet} \mathbb{F}$ is exact for every $R$-complex $\mathbb{G}$. If, moreover, $\mathbb{F}$ is a complex of projective $R$-modules, then it is null-homotopic (follows from Neeman's theorem).

We study the Fourier coefficients of generalized modular forms $f(\tau)$ of integral weight $k$ on subgroups $\Gamma$ of finite index in the modular group. We establish two Theorems asserting that $f(\tau)$ is constant if $k = 0$, $f(\tau)$ has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that $f(\tau)$ has a cuspidal divisor, $k$ is arbitrary, and $\Gamma = \Gamma_{0}(N)$, where we show that $f(\tau)$ is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

We answer a question posed in [12] on exponential integrability of functions of restricted $n$-energy. We use geometric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.

In this paper, we study a problem of extension of holomorphic functions given on a complex hypersurface with singularities on the boundary of a strictly pseudoconvex domain.