Let d be a square-free integer. Let

and {1, ω} forms a Z-basis for the ring of integers of the quadratic field We denote by Δ and hd the discriminant and the class number of respectively.

Dans l’article précédent [6], nous définissions deux classes des noyaux de convolution symétriques sur un groupe abélien localement compact et dénombrable à l’infini; l’une est régulière et l’autre est singulière.

Let L be a finite extension of a number field K with ring of integers and respectively. One can consider as a projective module over . The highest exterior power of as an module gives an element of the class group of , called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in is said to capitulate in L if its extension to is a principal ideal.)

J. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

It is a well-known result due to Sjogren [9] that if G is a finitely generated p-group then, for all n ≦ − 1, the (n + 2)-th dimension sub-group Dn+2(G) of G coincides with γn+2(G), the (n + 2)-th term of the lower central series of G. This was earlier proved by Moran [5] for n ≦ p − 2. For p = 2, Sjogren’s result is the best possible as Rips [8] has exhibited a finite 2-group G for which D4G) ≠ γ4(G) (see also Tahara [10, 11]). In this note we prove that if G is a finitely generated metabelian p-group then, for all . It follows, in particular, that, for p odd, for all n ≦ p and all metabelian p-groups G.

Throughout this paper, we denote by N, Q and R the set of all natural numbers containing 0, the set of all rational numbers, and the set of all real numbers, respectively.

Let R be a local ring with maximal ideal m and let us make the following definition according to the paper [NR] of Northcott and Rees, which is essentially due to F. S. Macaulay.

We will answer negatively to the question whether the completeness of infinitely sheeted covering surfaces of the extended complex plane have anything to do with their types being parabolic or hyperbolic. This will be accomplished by giving a one parameter family {W[α]: α ∈ A} of complete infinitely sheeted planes W[α] depending on the parameter set A of sequences α = (an)n>1 of real numbers 0 < an ≤ 1/2 (n ≥ 1) such that W[α] is parabolic for ‘small’ α’s and hyperbolic for ‘large’ α’s.

The homogenized ideal dual complex of an arbitrary meet-semilattice is introduced and described explicitly. Meet-distributive meet-semilattices whose homogenized ideal dual complex is level are characterized.

In the present work we describe 3-dimensional complex SL2-varieties where the generic SL2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional.

This is an ingredient of the classification [Keb99] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.

Let $K$ be a finitely generated extension of $\mathbb{Q}$, and let $A$ be a nonzero abelian variety over $K$. Let $\tilde{K}$ be the algebraic closure of $K$, and let $\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$ be the absolute Galois group of $K$ equipped with its Haar measure. For each $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let $\tilde{K}(\unicode[STIX]{x1D70E})$ be the fixed field of $\unicode[STIX]{x1D70E}$ in $\tilde{K}$. We prove that for almost all $\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers $l$ such that $A$ has a nonzero $\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order $l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

Let Ω(n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j = 0,1,…,m – 1, we have

with α = 1. Building on work of Kubota and Yoshida, we show that for m > 2 and any j = 0,1,…,m – 1, the error term is not o(xα) for any α < 1.

As an application of a sharp L2 extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

Let Ψ ∈ C2[0,1] be a positive real valued function on (0, 1]. Under certain assumptions on Ψ, the set is a pseudoconvex domain with C2-boundary which may be infinite type. If Ψ has flatness at 0 so that then we can obtain Lp(1 ≤ p ≤ ∞) estimates for on D.

We calculate extensions between certain irreducible admissible representations of p-adic groups.

In recent papers with S. S. Chern [3] and T.Ishihara [4], the author studied both the volume—and distance—decreasing properties of harmonic mappings thereby obtaining real analogues and generalizations of the classical Schwarz-Ahlfors lemma, as well as Liouville’s theorem and the little Picard theorem. The domain M in the first case was the open ball with the hyperbolic metric of constant negative curvature, and the target was a negatively curved Riemannian manifold with sectional curvature bounded away from zero. In this paper, it is shown that M may be taken to be any complete Riemannian manifold of non-positive curvature.

Previously the following question was offered by Zariski [6]:

Is any normal Noetherian local ring analytically irreducible?

In the present note, we will construct a counter-example against the question.

TERMINOLOGY. A ring (integrity domain) means always a commutative ring (integrity domain) with identity. A normal ring is an integrity domain which is integrally closed in its field of quotients. When 0 is an integrity domain, the integral closure of 0 in its field of quotients is called the derived normal ring of 0.

The content of Part I is nothing else than, the theory of binomial polynomial sequences in infinite variables (u(1), u(2), u(3), …) with weight u(1) = l. However, sometimes we are concerned with specialization therefore, we call the elements in K[u(1), u(2), u(3), …] differential polynomials. As analogies of special polynomials with binomial property, we may construct special differential polynomials with binomial property.