Totally positive extensions and weakly isotropic forms

The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.     

On weakly eutactic forms

We make precise some properties of the Hermite function in relationwith the Morse theory introduced by Avner Ash in his papers‘On eutactic forms’,Canad. J. Math. 29 (1977) 1040–1054and ‘On the existence of eutactic forms’,Bull. LondonMath. Soc. 12 (1980) 192–196, and with the cellular decompositionof the space of positive definite quadratic forms. We also establisha link between Ash's and Bavard's mass formulae.     

On positive definite hermitian forms

Kneser's method of constructing adjacent lattices will be used to determine class numbers of unimodular positive definite hermitian lattices of rank 2 and 3 over rings of integers in some imaginary quadratic fields. The same method will also be applied in order to construct indecomposable unimodular positive definite hermitian lattices of rank 2 and 3 over almost all orders in imaginary quadratic fields, even non-maximal ones. All exceptional cases will be determined explicitly. In part supported by NSF grant DMS 8805262. I would like to thank Prof. Martin Kneser for the support and the many helpful hints I received during the work on my Diplom thesis on which this paper is based, and also Prof. T.-Y. Lam for providing me with the grant without which I would not have been able to devote part of my time to writing this paper.     

On the zeros of weakly holomorphic modular forms

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen (Comment Math Univ St Pauli 52:55–57, 2003). Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.     

Minimal weakly isotropic forms

In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic subform of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways.     

On power means of positive quadratic forms

Some power means of positive definite quadratic forms are closely related to the fundamental scalar functions of the matrix associated with the quadratic form. This relation can (among other things) be used to give new proofs of some of the classical matrix inequalities.     

Generalized Serre relations for Lie algebras associated with positive unit forms

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre’s Theorem [J.-P. Serre, Complex Semisimple Lie Algebras (G.A. Jones, Trans.), Springer-Verlag, New York, 1987] gives then a representation of the given Lie algebra in generators and relations in terms of the Cartan matrix.In this work, we generalize Serre’s Theorem to give an explicit representation in generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of linearly independent roots which generate all roots as linear combinations with integral coefficients.     

On positive definite quadratic forms in correlatedt variables

Summary In this paper we extend Ruben's [4] result for quadratic forms in normal variables. He represented the distribution function of the quadratic form in normal variables as an infinite mixture of chi-square distribution functions. In the central case, we show that the distribution function of a quadratic form int-variables can be represented as a mixture of beta distribution functions. In the noncentral case, the distribution function presented is an infinite series in beta distribution functions. An application to quadratic discrimination is given.     

On Voronoi reduction of positive definite quadratic forms

There are two methods of reduction of positive definite quadratic forms due to Voronoi. One of these methods is based on the perfect forms and the other on the type of Voronoi polyhedra associated with the form. It was conjectured by Voronoi that these two methods are strongly connected.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i ₁, . . . , i _l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l −1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function K^(l)_I(R){K^{(l)}_I(R)} which counts the number of tuples (i ₁, . . . , i _l ) such that l_i1,?,l_il ￡ R²{\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2} and l_ij-l_i1 ? I_j{\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j} for 2 ≤ j ≤ l. We study the asymptotic behavior of K^(l)_I(R){K^{(l)}_I(R)} as R tends to infinity. If k ≥ 4 we prove K^(l)_I(R) ~ c_l(Q) vol(I)R^lk-2(l-1){K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}} for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

On nondecomposable positive definite Hermitian forms over imaginary quadratic fields

Methods are presented for the construction of nondecomposable positive definite integral Hermitian forms over the ring of integers R_m of an imaginary quadratic field ℚ(√−m). Using our methods, one can construct explicitly an n-ary nondecomposable positive definite Hermitian R_m-lattice ( L, h) with given discriminant 2 for every n⩾2 (resp. n⩾13 or odd n⩾3) and square-free m = 12 k + t with k⩾1 and t∈ (1,7) (resp. k⩾1 and t = 2 or k⩾0 and t∈ 5,10,11). We study also the case for discriminant different from 2.