In this paper, we use the theta function method [10] to give explicit Doi-Naganuma type maps associated to an imaginary quadratic field K, lifting cusp forms on any congruence subgroup of SL(2, Z) to forms on SL(2, C) automorphic with respect to an appropriate arithmetic discrete subgroup. The case of class number one, and form modular with respect to group Γ0(D) and character χ0 = (–D/*), where –D is the discriminant of K has been treated by Asai [1]. In order to complete his discussion, we must first introduce a more general theta function associated to an indefinite quadratic form (here of type (3, 1)), which we regard as a specialization of a symplectic theta function (see also [4]).

The purpose of this paper is to establish the following isomorphism of Lie algebras.

Main Theorem. Let X be the complement of a hypersurface S in the complex projective space PN. Then the tower of nilpotent complex Lie algebras associated with the fundamental group π1(X, *) and the holonomy Lie algebraattached to S are isomorphic.

A pair (X, D) of a projective variety X and a very ample divisor D on X is called stable (resp. semi-stable, resp. unstable) if the Chow point corresponding to the embedding is SL(N + 1)-stable (resp. semi-stable, resp. unstable). The criterion for stability is one of the most important steps in proving the existence of moduli spaces.

Let Ω be a bounded domain in Rn with coordinates x = (x1, …, xn) such that Ω contains the origin. We consider the elliptic operators

where are real valued and in C2(Ω).

Let A be a Noetherian local ring and p a prime ideal in A. Let

and call them, respectively, the Rees ring and the form ring of p. The purpose of this paper is to clarify, provided p is an almost complete intersection in A (cf. (2.1) for definition), when the rings R(p) and G(p) are Gorenstein.

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.

This paper includes a proof, relative to the consistency of ZFC, of the consistency of ZFC, the continuum has singular cardinality and the extended plus-one hypothesis.

Let (R, m) be a local noetherian ring and I a proper ideal in R. Let (I) be the Rees-ring ⊕n≥0 In with respect to I. In this note we describe conditions for I and R in order that the Cohen-Macaulay property (C-M for short) of R/I can be lifted to R and (I), see Propositions 1.2, 1.3. and 1.4.

Let Rn be the Euclidean space of dimension n ≧ 1 with the standard inner product and the norm be the unit sphere {x ∈ Rn; |x| = 1} and dωn-1 be the volume element of Sn-1 such that Sn-1 gets the volume 1. Let Ω be an open set of Rn containing Sn-1 and let f: Ω → Rm be a smooth map.

Let L, M be regular quadratic lattices over Zp. The local density αp(L, M) is an important invariant in the theory of representation of quadratic forms and they appear naturally in Fourier coefficients of Eisenstein series. In spite of the importance we knew little except the case when either L = M or rk L = 1 and M is unimodular. Evaluating them is a laborious task.

Let X be a locally compact and σ-compact abelian group and be the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure on associated with ξ.

The purpose of this paper is to show that the corona theorem is valid on any unbounded finitely sheeted covering surface if and only if the corona theorem is valid on its base surface.

It is well known that the quasi-exellent property (i.e. G-ring and J-II) ascends by finite type morphisms. On the other hand Greco showed in [2, Proposition 4.1 and Proposition 2.3] that G-property (i.e. formal fibers are geometrically regular) does not descend by finite type morphisms, although J-II property (i.e. any finite type algebra has regular locus open) does.

Let k be a field of characteristic 0, let be an algebraic closure of k and denote by Gk = G(/k) the absolute Galois group of k. Suppose that for some natural number n ≥ 3 the cohomology group Hn(Gk) Z) is trivial.