On Siegel’s product formulas for quadratic forms over algebraic number fields in higher dimensional cases

For a totally real algebraic number field k, it is known that every (partial) zeta function of k is a finite sum of Dirichlet series which are regarded as natural generalizations of the Hurwits zeta function (see [1] and [2]). In this note we show that the similar result holds for arbitrary (not necessarily totally real) algebraic number field. At the time of the Bombay Colloquium (1979), H. M. Stark orally communicated to the author that he has obtained such a result for non-real cubic fields. His oral communication was an initial impetus to the present work. The author wishes to express his gratitude to Stark.

The extensive study of residue classes to given moduli which are represented by a quadratic form leads to an application to sequences in algebraic number fields. So-called integer-detecting sequences were introduced over the rationals in earlier work. We discuss the respective concept for sequences in number fields. 1. Introduction and definitions In this paper, we study quadratic forms x 2 + Dy 2 mod s for given moduli s and try to decide which residue classes are represented by them. Quadratic forms of this type may be considered as norms in quadratic number fields. These norms and, more generally, norms in arbitrary number fields occur naturally when studying certain sequences, so-called integer-detecting sequences. We first shall explain this notion for rational numbers. Integer-detecting sequences were introduced in [12] (see also [3]), and a bunch of necessary or sufficient criteria was derived. For the convenience of the reader let us review the definition: A sequence (mn) n 0 of positive integers with mn 0) if any rational number r ,s atisfy� � .

On the Fourier coefficients of Hilbert–Maass wave forms of half integral weight over arbitrary algebraic number fields

On the period length of the generalized Lagrange algorithm

On the modular forms of weight 1/2 over algebraic number fields

Like the previous chapter this one deals with the number theoretic aspect of the theory of quadratic forms. Instead of the integers \( \mathbb{Z} \) and the rational field \( \mathbb{Q} \) we consider more generally an arbitrary algebraic number field and its ring of algebraic integers. Now and then we can be even more general by considering the quotient field of an arbitrary Dedekind domain. The theory of quadratic forms over algebraic number fields and the corresponding rings of integers is developed in detail in several books, especially O’Meara [1963] (to be referred to by OM). However, in the last few years several interesting results have been added to this theory. These new results, particularly those concerning the calculation of the Witt group are emphasized in this chapter.KeywordsSymmetric Bilinear FormGlobal FieldDiscrete Valuation RingDedekind DomainQuotient FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields. For the most part, we work over an arbitrary number field rather than the rationals. The first half of the article is expository and is intended to be accessible to graduate students and advanced undergraduates. It gives a self-contained treatment, following Schur, over the field of rational numbers.

On Functional Equations of Zeta Distributions

?1. Let (E be the matrix of a quadratic form with rational integral coefficients and F (e) the group of integral solutions U of 25 [U] = U'2U = S. The group F (e) is called the unit group of (E and its elements the units of S. Eisenstein defined the measure of F (5), when (E is definite, as the reciprocal of the order of F (S). The generalization to arbitrary (E is due to C. L. Siegel. It is the object of this note to obtain a generalization of Siegel's results to the case where (a and 11 have integral elements in an arbitrary algebraic number field K. Since the results are useful in furnislhing examples of discontinuous groups in higher complex spaces (see [1] and [61), we formulate and prove our results from the standpoint of the theory of discontinuous groups. From this standpoint, it appears that with every indefinite quadratic form with integral elements in K, there is associated a discontinuous group of mappings leaving invariant an open bounded subspace of a certain real Euclidean space. This subspace depends only on the signature of the quadratic form. Our main theorem asserts that except for a certain trivial case, this discontinuous group isomorphic to the group of units of e is, in the terminology of Siegel, a group of the first kind. The special case when K has no complex infinite prime spot, was studied by us recently. The units of quadratic forms with algebraic coefficients were studied also in a few special cases by Dirichlet, Fricke and others. In a future paper we shall apply our results to the analytic theory of indefinite quadratic forms over algebraic number fields.'

For an arbitrary (not totally real) number field K of degree ≥3, we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ(τ(γ p ))≤X for each embedding τ of K into the complex field. (X a large real parameter, p≥2 a fixed integer, and μ(z)=max(| Re (z)|,| Im (z)|) for any complex z.) This quantity is evaluated asymptotically in the form c p,K X n/p +R p,K (X), with sharp estimates for the remainder R p,K (X). The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.

The investigation of nonsingular bilinear forms originates from the classification of division algebras over the real number field. Building upon this foundation, researchers have delved into the study of nonsingular bilinear forms over real number fields, leading to significant results such as Hopfs theorem. However, the interest in understanding nonsingular bilinear forms extends beyond real number fields, prompting a desire to explore other fields as well. When it comes to algebraically closed fields, the theorem becomes well-understood, with essence captured by the Hopf-Smith theorem. Inspired by these established studies, we are motivated to further the comprehension of nonsingular bilinear forms over arbitrary fields. Given a field , we study in this article numerical constraints on for the existence of nonsingular bilinear maps for not only algebraically closed fields and the real number field but also the rational number field and finite fields. We reach the final conclusion mainly through algebro-geometric techniques and the use of determinantal varieties. We reprove a result of HopfSmith which states that the minimal possible value of is when is an algebraically closed field. When is the real number field, we prove that under a combinatorial condition, the minimal possible value of is still . We also show that when is the rational number field or a finite field, the minimal possible value of is .

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

For a certain sum of radicals the author presents a Monte Carlo algorithm that runs in polynomial time to decide whether the sum is contained in some number field Q( alpha ), and, if so, its coefficient representation in Q( alpha ) is computed. As a special case the algorithm decides whether the sum is zero. The main algorithm is based on a subalgorithm which is of interest in its own right. This algorithm uses probabilistic methods to check for an element beta of an arbitrary (not necessarily) real algebraic number field Q( alpha ) and some positive rational integer r whether there exists an rth root of beta in Q( alpha ). >

We consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of real algebraic numbers. In both the discounted and the limiting average versions of these games, we prove that the value vector also lies in the same field of real algebraic numbers. Our method supplies finite construction of univariate polynomials whose roots contain these value vectors. In the case where the data of the game are rational, the method also provides a way of checking whether the entries of the value vectors are also rational.

The Dirichlet class number formula expresses the residue at s = 1 of the Dedekind zeta function ζ F(s) of an arbitrary algebraic number field F as the product of a simple factor (involving the class number of the field) with the determinant of a matrix whose entries are logarithms of units in the field. On the other hand, if F is a totally real number field of degree n, then a famous theorem by Klingen and Siegel says that the value ζ F (m) for every positive even integer in is a rational multiple of π mn In [52] and [53], a conjectural generalization of these two results was formulated according to which the special value ζ F (m) for arbitrary number fields F and positive integers m can be expressed in terms of special values of a transcendental function depending only on m, namely the m th classical polylogarithm function. These instances are expected to form part of a much more general picture in which a special value of an L-series of “motivic origin” is expressed in terms of some transcendental function. In this survey we collect some pieces fitting into and illustrating this picture.