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2013 | Buch

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Quadratic and Higher Degree Forms

herausgegeben von: Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep

Verlag: Springer New York

Buchreihe : Developments in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In the last decade, the areas of quadratic and higher degree forms have witnessed dramatic advances. This volume is an outgrowth of three seminal conferences on these topics held in 2009, two at the University of Florida and one at the Arizona Winter School. The volume also includes papers from the two focused weeks on quadratic forms and integral lattices at the University of Florida in 2010.Topics discussed include the links between quadratic forms and automorphic forms, representation of integers and forms by quadratic forms, connections between quadratic forms and lattices, and algorithms for quaternion algebras and quadratic forms. The book will be of interest to graduate students and mathematicians wishing to study quadratic and higher degree forms, as well as to established researchers in these areas.

Quadratic and Higher Degree Forms contains research and semi-expository papers that stem from the presentations at conferences at the University of Florida as well as survey lectures on quadratic forms based on the instructional workshop for graduate students held at the Arizona Winter School. The survey papers in the volume provide an excellent introduction to various aspects of the theory of quadratic forms starting from the basic concepts and provide a glimpse of some of the exciting questions currently being investigated. The research and expository papers present the latest advances on quadratic and higher degree forms and their connections with various branches of mathematics.

Inhaltsverzeichnis

Frontmatter

Toy Models for D. H. Lehmer’s Conjecture II

Abstract

In the previous paper under the same title, we showed that the m-th Fourier coefficient of the weighted theta series of the ${\mathbb{Z}}^{2}$-lattice and the A ₂-lattice does not vanish when the shell of norm m of those lattices is not the empty set. In other words, the spherical 4 (resp. 6)-design does not exist among the nonempty shells in the ${\mathbb{Z}}^{2}$-lattice (resp. A ₂-lattice). This paper is the sequel to the previous paper. We take 2-dimensional lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, then, show that the m-th Fourier coefficient of the weighted theta series of those lattices does not vanish, when the shell of norm m of those lattices is not the empty set. Equivalently, we show that the corresponding spherical 2-design does not exist among the nonempty shells in those lattices.

Eiichi Bannai, Tsuyoshi Miezaki

On Representation of an Integer by X 2 + Y 2 + Z 2 and the Modular Equations of Degree 3 and 5

Abstract

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that

$$\displaystyle{ s(25n) = \left (6 -\left (-n\vert 5\right)\right)s(n) - 5s\left (\frac{n} {25}\right) }$$

follows easily from the well known Ramanujan modular equation of degree 5. Moreover, I establish new relations between s(n) and h(n), g(n), the number of representations of n by the ternary quadratic forms

$$\displaystyle{2{x}^{2} + 2{y}^{2} + 2{z}^{2} - yz + zx + xy,\quad {x}^{2} + {y}^{2} + 3{z}^{2} + xy,}$$

respectively. Finally, I propose a remarkable new identity for s(p ² n)−p s(n) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p ², 16p ².

Alexander Berkovich

Almost Universal Ternary Sums of Squares and Triangular Numbers

Abstract

For any integer x, let T _x denote the triangular number $\frac{x(x+1)} {2}$. In this paper we give a complete characterization of all the triples of positive integers (α, β, γ) for which the ternary sums $\alpha {x}^{2} +\beta T_{y} +\gamma T_{z}$ represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun (Trans Am Math Soc 362:6425–6455, 2010, Conjecture 1.19(i)) and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.

Wai Kiu Chan, Anna Haensch

Weighted Generating Functions for Type II Lattices and Codes

Abstract

We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of Euclidean lattices. Namely, we use the finite-dimensional representation theory of $\mathfrak{s}\mathfrak{l}_{2}$ to derive a decomposition theorem for the spaces of discrete homogeneous polynomials in terms of the spaces of discrete harmonic polynomials, and prove a generalized MacWilliams identity for harmonic weight enumerators. We then present several applications of harmonic weight enumerators, corresponding to some uses of weighted theta functions: an equivalent characterization of t-designs, the Assmus–Mattson Theorem in the case of extremal Type II codes, and configuration results for extremal Type II codes of lengths 8, 24, 32, 48, 56, 72, and 96.

Noam D. Elkies, Scott Duke Kominers

Quadratic Forms and Automorphic Forms

Abstract

These notes give a friendly four-part introduction to various aspects of the arithmetic and analytic theories of quadratic forms, aimed at a graduate-level audience. The main themes discussed are: geometry and local-global methods, theta functions and Siegel’s theorem, Clifford algebras and spin groups, and adelic theta liftings via the Weil representation.

Jonathan Hanke

Integral Positive Ternary Quadratic Forms

Abstract

We discuss some families of integral positive ternary quadratic forms. Our main example is $f(x,y,z) = {x}^{2} + {y}^{2} + 16n{z}^{2},$ where n is positive, squarefree, and $n = {u}^{2} + {v}^{2}$ with $u,v \in \mathbf{Z}.$

William C. Jagy

Some Aspects of the Algebraic Theory of Quadratic Forms

Abstract

This article, based on the lectures at the Arizona Winter School on “Quadratic forms”, gives a quick introduction to the algebraic theory of quadratic forms. It discusses some invariants associated to quadratic forms like the Pythagoras number and the u-invariant and touches on some recent progress on these topics.

R. Parimala

On the Length of Binary Forms

Abstract

The K-length of a form f in $K[x_{1},\ldots,x_{n}]$, $K \subset \mathbb{C}$, is the smallest number of d-th powers of linear forms of which f is a K-linear combination. We present many results, old and new, about K-length, mainly for n = 2, and often about the length of the same form over different fields. For example, the K-length of $3{x}^{5} - 20{x}^{3}{y}^{2} + 10x{y}^{4}$ is three for $K = \mathbb{Q}(\sqrt{-1})$, four for $K = \mathbb{Q}(\sqrt{-2})$ and five for $K = \mathbb{R}$.

Bruce Reznick

Representation of Quadratic Forms by Integral Quadratic Forms

Abstract

The number of representations of a positive definite integral quadratic form of rank n by another positive definite integral quadratic form of rank m ≥ n has been studied by arithmetic, analytic, and ergodic methods. We survey and compare in this article the results obtained by these methods.

Rainer Schulze-Pillot

Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

Abstract

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M₂(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.

John Voight

Titel: Quadratic and Higher Degree Forms
herausgegeben von: Krishnaswami Alladi
Manjul Bhargava
David Savitt
Pham Huu Tiep
Verlag: Springer New York
Electronic ISBN: 978-1-4614-7488-3
Print ISBN: 978-1-4614-7487-6
DOI: https://doi.org/10.1007/978-1-4614-7488-3