Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds:

where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).

Canonical and terminal singularities are introduced by M. Reid [5], [6]. He proved that 3-dimensional terminal singularities are cyclic quotient of smooth points or cDV points [6].

The relationship between Poisson’s summation formula and Hamburger’s theorem [2] which is a characterization of Riemann’s zetafunction by the functional equation was already mentioned in Ehrenpreis-Kawai [1]. There Poisson’s summation formula was obtained by the functional equation of Riemann’s zetafunction. This procedure is another proof of Hamburger’s theorem. Being interpreted in this way, Hamburger’s theorem admits various interesting generalizations, one of which is to derive, from the functional equations of the zetafunctions with Grössencharacters of the Gaussian field, Poisson’s summation formula corresponding to its ring of integers [1], The main purpose of the present paper is to give a generalization of Hamburger’s theorem to some zetafunctions with Grössencharacters in algebraic number fields. More precisely, we first define the zetafunctions with Grössencharacters corresponding to a lattice in a vector space, and show that Poisson’s summation formula yields the functional equations of them. Next, we derive Poisson’s summation formula corresponding to the lattice from the functional equations.

We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.

The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

Let τ denote a point of the Siegel upper-half space be a (column) vector in Cg and m a vector with m′ resp. m″ in Zg as its first resp. second entry vectors, then the series

in which , represents an holomorphic function on the product and it is called the theta function of characteristic m.

In this paper, we study higher reciprocity law of irreducible polynomials f(x) over Q of degree 3, especially, its close connection with elliptic curves rational over Q and cusp forms of weight 1. These topics were already studied separately in a special example by Chowla-Cowles [1] and Hiramatsu [2]. Here we bring these objects into unity.

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

Let G be a connected linear algebraic group, p a rational representation of G on a finite-dimensional vector space V, all defined over C.