In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

Recent development in hypergeometric functions in several variables has made the importance of studying twisted rational de Rham cohomology clear to many specialists. Roughly speaking, a hypergeometric function in our sense is the integral of a product of complex powers of polynomials Pj(u1, . . . . ,un) : ∫ U du1 ∧ · · · ∧ dun, U = Π , integration being taken over some cycle. So we are led naturally to consider the twisted rational de Rham cohomology, which is a direct generalization of the usual de Rham cohomology to multivalued case.

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

Let G denote a locally compact commutative group with a lattice Γ, G* its dual, and for every (g, g*) in G × G*; let Γ* denote the annihilator of Γ in G* and dg the Haar measure on G such that G/Γ is of measure 1.

Dans toute la suite X désigne un groupe abélien localement compact et dénombrable à l’infini; ξ est sa mesure de Haar. Nous rappelons qu’un noyau de convolution N sur X signifie une mesure de Radon positive dans X. Il est connu que le principe de domination pour N joue un grand rôle dans la théorie du potentiel. Nous remarquons ici qu’il existe deux sortes des principes de domination pour N; l’un est défini par G. Choquet et J. Deny (cf. [3]), qui est très utile pour discuter le principe du balayage, et l’autre est introduit de celui dans le cardre plus large (cf. [8]). Nous montrerons d’abord que ceux sont équivalents. Par conséquent, on pourra discuter, en même temps, sur les mesures balayées relativement au noyau N et sur la résolvante associée au noyau N.

En Géométrie Algébrique on a un critère pour qu’une surface moins une courbe soit affiine (Hartshorne (5)). Dans (5), on demande s’il existe un analogue analytique. Le but de cet article est de donner une condition numérique necessaire pour les surfaces complexes compactes (Théorème 1) et une condition suffisante pour les surfaces réglées (Théorème 2).

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

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

A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition. R is called a left QF-3 ring if it contains a faithful projective injective left ideal. If R is semi-primary and left QF-3, then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left R-module [5], in agreement with the definition of QF-3 algebra given by R.M. Thrall [6]. Let Q(M) denote the injective envelope of a (left) R-module M. We call R left QF-3+ if Q(R) is projective. J.P. Jans showed that among rings with minimum condition on left ideals, the classes of QF-3 and QF-3+ rings coincide [5].

By extending Mumford’s result on the generating by global sections there are estimates on the degree for generators of local cohomology modules. These arguments provide bounds on the Castelnuovo-Mumford regularity, in particular for Cohen-Macaulay varieties. As an application they imply a few more cases of varieties that satisfy a conjecture posed by Eisenbud and Gôto.

Consider the complex of differential forms on an open affine subvariety U of AN with differential where d is the usual exterior derivative and ø is a fixed 1-form on U. For certain U and ø, we compute the cohomology of this complex.

In this paper, we shall consider the birational structure of moduli of stable sheaves on elliptic surfaces, which is a generalization of Friedman’s results to higher rank cases. As applications, we show that some moduli spaces of stable sheaves on ℙ2 are rational. We also compute the Picard groups of those on Abelian surfaces.

The purpose of the present note is to reform Mr. K. Nomizu’s result on the group of all affine transformations of an affinely connected manifold. We shall prove the following.

In this article, we present a concavity property of the minimal $L^{2}$ integrals related to multiplier ideal sheaves with Lebesgue measurable gain. As applications, we give necessary conditions for our concavity degenerating to linearity, characterizations for 1-dimensional case, and a characterization for the holding of the equality in optimal $L^2$ extension problem on open Riemann surfaces with weights may not be subharmonic.

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.

Let R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions.

Soient X un groupe abélien localement compact et non-compact, et dx sa mesure de Haar. Rappelons qu’un noyau de convolution N sur X est une mesure de Radon positive dans X. Il est symétrique s’il est symétrique par rapport à l’origine. Notons MK la totalité de fonctions mesurables, bornées dans X, à valeurs réelles et à support compact. On dit que N est de type positif si, quelle que soit f une fonction de et la signe * est la convolution sur X.