The representation of quadratic forms by almost universal forms of higher rank

 In this article, we prove that there are only finitely many positive definite integral quadratic forms of rank n+3(n≥2) that represent all positive definite integral quadratic forms of rank n but finitely many exceptions. Furthermore we determine all diagonal quadratic forms having such property and its exceptions remaining four as candidates. Received: 29 November 2000 ; in final form: 8 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 11E12, 11E20.     

Absolute stable rank and quadratic forms over noncommutative rings

Given an associative ring A, let asr(A) denote the absolute stable range of A, as defined in [5]. We prove that asr(A) 1 + Kdim A if A is a right Noetherian ring, and that asr(A) 1 + cl-Kdim A if A is an affine PI algebra. Combined with results from [5], this provides a cancellation theorem ('Witt cancellation') for quadratic spaces defined over such a ring A.     

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x ²+Ny ² for a squarefree integer N.      

Discriminant algebras of finite rank algebras and quadratic trace modules

Based on the construction of the discriminant algebra of an even-ranked quadratic form and Rost’s method of shifting quadratic algebras, we give an explicit rational construction of the discriminant algebra of finite-rank algebras and, more generally, of quadratic trace modules, over arbitrary commutative rings. The discriminant algebra is a tensor functor with values in quadratic algebras, and a symmetric tensor functor with values in quadratic algebras with parity. The automorphism group of a separable quadratic trace module is a smooth, but in general not reductive, group scheme admitting a Dickson type homomorphism into the constant group scheme Z ₂.     

Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

Cobordisms of finite quadratic forms and gluing oriented manifolds

We develop certain discriminant form techniques allowing one to “glue” two integral lattices into a new lattice. In particular, we introduce the notion of fusion of finite forms, which is closely connected with the more familiar notion of cobordism of such forms. We also consider the category of finite forms and their cobordisms. These techniques are used for describing the intersection form of a compact oriented 4k-manifold M obtained by gluing together two manifolds M₁ and M₂ along one or several boundary components (the intersection forms of M₁ and M₂ are assumed to be nondegenerate). Bibliography: 12 titles. Translated fromZapiski nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 115–132.     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.

     

Multiplicities of non-arithmetic ternary quadratic forms and elliptic curves of positive rank

In this note we will characterize all non-arithmetic ternary quadratic forms with unbounded multiplicities. It turns out that their existence is closely related with certain classes of elliptic curves whose investigation is interesting on its own. Received March 3, 1997; in final form November 13, 1997     

Representations of integers by binary forms and the rank of the Mordell-Weil group

Inventiones mathematicae -     

Representation of numbers by quadratic forms in algebraic number fields

Let K be the totally real field of algebraic numbers of degree n=[k2e] with the discriminant D=D(K); t=t(x₁, ..., x_s) a totally positive quadratic form of the determinant d>0 over the ring of integers from the field K; S4. Let be the number of representations over of the number m by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m with m totally positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 68–77, 1986.     

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

     

Quantum Jacobi forms and finite evaluations of unimodal rank generating functions

In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expression for radial limits associated to the partition rank and crank functions previously studied by Ono, Rhoades, and Folsom.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.