Let ${\mathfrak g}$ be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with ${\mathfrak h}$ the Cartan subalgebra and $Q \subset {\mathfrak h}^{*}$ the root lattice. Denote by $\Theta_{Q}(q)$ the theta series of the root lattice $Q$ of ${\mathfrak g}$. We prove a curious "combinatorial" identity for the derivative of $\Theta_{Q}(q)$, i.e.\ for $q \frac{d}{dq} \Theta_{Q}(q)$, by using the representation theory of an affine Lie algebra.

To a polynomial $f$ over a non-archimedean local field $K$ and a character $\chi$ of the group of units of the valuation ring of $K$ one associates Igusa's local zeta function $Z(s, f, \chi)$. In this paper, we study the local zeta function $Z(s, f, \chi)$ associated to a non-degenerate polynomial $f$, by using an approach based on the $p$-adic stationary phase formula and Néron $p$-desingularization. We give a small set of candidates for the poles of $Z(s, f, \chi)$ in terms of the Newton polyhedron $\Gamma(f)$ of $f$. We also show that for almost all $\chi$, the local zeta function $Z(s, f, \chi)$ is a polynomial in $q^{-s}$ whose degree is bounded by a constant independent of $\chi$. Our second result is a description of the largest pole of $Z(s, f, \chi_{triv})$ in terms of $\Gamma(f)$ when the distance between $\Gamma(f)$ and the origin is at most one.

The present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of $n$-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Let $D$ be a bounded strictly pseudoconvex domain in ${\mathbb C}^{n}$ (with not necessarily smooth boundary) and let $X$ be a submanifold in a neighborhood of $\overline{D}$. Then any $L^{p}$ $(1 \leq p < \infty)$ holomorphic function in $X \cap D$ can be extended to an $L^{p}$ holomorphic function in $D$.

The main purpose of this paper is to prove the Iitaka conjecture $C_{n, m}$ on the assumption that the sufficiently general fibers have maximal Albanese dimension.

We investigate period maps of polarized variations of Hodge structures of weight one or two. We treat the case when the period domains are bounded symmetric domains. We deal with a relationship between canonical extensions of some Hodge bundles and automorphic forms. As applications, we obtain a canonical bundle formula for certain algebraic fiber spaces, such as Abelian fibrations, $K3$ fibrations, and solve Iitaka's famous conjecture $C_{n, m}$ for some algebraic fiber spaces.

The new concept of a Liouville sequence is introduced in this paper by mean of the related Liouville series. Main results are two criteria for when certain sequences are Liouville. Several applications are presented. A counterexample is included for the case that we substantially weaken the hypotheses in the main results.