Let $R$ be a commutative Noetherian local ring. This paper deals with the problem asking whether $R$ is Gorenstein if the $n$th syzygy module of the residue class field of $R$ has a non-trivial direct summand of finite G-dimension for some $n$. It is proved that if $n$ is at most two then it is true, and moreover, the structure of the ring $R$ is determined essentially uniquely.

We prove an analog of the classical Hartogs extension theorem for $CR$ $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a question formulated in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds.

We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.

We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ([6]), where we developed the analytic theory of log abelian varieties.

Let $K/F$ be a quadratic extension of number fields. After developing a theory of the Eisenstein series over $F$, we prove a formula which expresses a partial zeta function of $K$ as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the $0$-th Laurent coefficients at $s=1$ of zeta functions of $K$ and $F$.

In this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.