Theory Of Integral Quadratic Forms Research Articles

Effective Estimates on Integral Quadratic Forms: Masser’s Conjecture, Generators of Orthogonal Groups, and Bounds in Reduction Theory

In this paper we prove a conjecture of David Masser on small height integral equivalence between integral quadratic forms. Using our resolution of Masser’s conjecture we show that integral orthogonal groups are generated by small elements which is essentially an effective version of Siegel’s theorem on the finite generation of these groups. We also obtain new estimates on reduction theory and representation theory of integral quadratic forms. Our line of attack is to make and exploit the connections between certain problems about quadratic forms and group actions, whence we may study the problem in the well-developed theory of homogeneous dynamics, arithmetic groups, and the spectral theory of automorphic forms.

Bulk-edge correspondence in (2 + 1)-dimensional Abelian topological phases

The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at $\nu=8$ and 12, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\nu=8/7, 12/11, 8/15, 16/5$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a $p^n$-th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $U(1)^N$ Chern-Simons theory parameterized by a K-matrix.

Zeta-functions of harmonic theta-series and prime numbers

Speaking on rational prime numbers in various arithmetical sequences, it may be noted that no essential progress have been achieved for more than century and a half since famous Dirichlet theorem on prime numbers in arithmetic progressions (1837). Absolutely mystical is still the question on prime numbers in quadratic sequences, i.e., on prime numbers of the form an+bn+c, where a, b, c are rational coprime integers, and d = b − 4ac is not a rational square. The situation is not changing despite of considerable progress of the algebraic-analytical theory of integral quadratic forms reached after Dirichlet. Our purpose here is to draw the attention of numbertheorist to some analytical aspects of the theory of quadratic forms possibly related to the problem. In order to be more concrete, let us start from the the celebrated problem on prime numbers of the form 1+n. It is well known that the problem is closely related to reductions prime modules of certain elliptic curves with complex multiplications by Gauss integers a+ √ −1b, say, the curve y = x(x − 1). (1.1)

Integral quadratic forms, Kac-Moody algebras, and fractional quantum Hall effect. AnADE-O classification

The purpose of this paper is to present a rather comprehensive classification of incompressible quantum Hall states in the limit of large distance scales and low frequencies. In this limit, the description of low-energy excitations above the groundstate of an incompressible quantum Hall fluid is intimately connected to the theory of integral quadratic forms on certain lattices which we call quantum Hall lattices. This connection is understood with the help of the representation theory of algebras of gapless, chiral edge currents or, alternatively, from the point of view of the bulk effective Chern-Simons theory. Our main results concern the classification of quantum Hall lattices in terms of certain invariants and their enumeration in low dimensions and for a limited range of values of those invariants. Among physical consequences of our analysis we find explicit, discrete sets of plateau values of the Hall conductivity, as well as the quantum numbers of quasiparticles in fluids corresponding to any one among those quantum Hall lattices. Furthermore, we are able to predict transitions between structurally different quantum Hall fluids corresponding to the same filling factor. Our general results are illustrated by explicitly considering the following plateau values: σ H =N/(2N±1),N=1, 2, 3,..., σ N =1/2.

Theta series of ternary and quaternary quadratic forms

In the theory of integral quadratic forms or in the allied subject of modular forms very little seems to be known about the linear independence of theta series associated to positive definite forms. For a system of inequivalent positive binary quadratic forms with a fixed fundamental discriminant, Hecke [4-1 showed that their theta series are independent. However, for positive forms in three or more variables, even in a fixed genus, there does not appear to be any non-trivial general result of a similar nature in that certain explicitly described forms in the genus yield linearly independent theta series. We consider here even positive definite ternary quadratic forms of discriminant 2p and quaternary quadratic forms of discriminant p where p is a prime congruent to 1 mod4. For a fixed such p these forms belong to a single genus G'.=G(3,2p), resp. G(4,p). Let G' denote the subset consisting of those forms in G which represent 2. In [6] Kitaoka observed that the number of form-classes in G'(4, p)

Partitionings of a genus of quadratic forms

A description of the quasigenus, which was introduced into the theory of integral quadratic forms by B. W. Jones, is given in terms of the general theory of spinor general of lattices over Dedekind domains. As a consequence several questions raised by Jones regarding the relationship between the partitionings of a genus induced by the splittings into spinor genera and quasigenera are answered.

Worst-case complexity bounds for algorithms in the theory of integral quadratic forms

The problem of factoring an integer and many other number-theoretic problems can be formulated in terms of binary quadratic Diophantine equations. This class of equations is also significant in complexity theory, subclasses of it having provided most of the natural examples of problems apparently intermediate in difficulty between P and NP-complete problems, as well as NP-complete problems [2, 3, 22, 26]. The theory of integral quadratic forms developed by Gauss gives some of the deepest known insights into the structure of classes of binary quadratic Diophantine equations. This paper establishes explicit polynomial worst-case running time bounds for algorithms to solve certain of the problems in this theory. These include algorithms to do the following: (1) reduce a given integral binary quadratic form; (2) quasi-reduce a given integral ternary quadratic form; (3) produce a form composed of two given integral binary quadratic forms; (4) calculate genus characters of a given integral binary quadratic form, when a complete prime factorization of its determinant D is given as input; (5) produce a form that is the square root under composition of a given form (when it exists), when a complete factorization of D and a quadratic nonresidue for each prime dividing D is given as input.

Some examples of integral definite quaternary quadratic forms with prime discriminant

In the theory of integral quadratic forms or the related theory such as modular forms, it is important to give examples of forms with given discriminant for various computational purposes. However, little is known about this problem even in the case where the explicit formula for the class number has been given. Among the works that deal with satisfactorily many examples or give a constructive method of them, there are [1] for ternary forms, and [5] for quaternary forms with square discriminants. As for quaternary forms with prime discriminant, P. Ponomarev [8] gave an explicit way to construct such forms from ternary forms with discriminant 2q. It was based on the result of his previous work [7] and that of Y. Kitaoka [6], but was rather complicated.