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Book DetailsContemporary MathematicsVolume: 587; 2013; 243 ppMSC: Primary 11;
This volume contains the proceedings of the International Workshop on Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, held November 13–18, 2011, at the Banff International Research Station, Banff, Alberta, Canada.
The articles in this volume cover the arithmetic theory of quadratic forms and lattices, as well as the effective Diophantine analysis with height functions. Diophantine methods with the use of heights are usually based on geometry of numbers and ideas from lattice theory. The target of these methods often lies in the realm of quadratic forms theory. There are a variety of prominent research directions that lie at the intersection of these areas, a few of them presented in this volume: Representation problems for quadratic forms and lattices over global fields and rings, including counting representations of bounded height.
 Small zeros (with respect to height) of individual linear, quadratic, and cubic forms, originating in the work of Cassels and Siegel, and related Diophantine problems with the use of heights.
 Hermite's constant, geometry of numbers, explicit reduction theory of definite and indefinite quadratic forms, and various generalizations.
 Extremal lattice theory and spherical designs.
ReadershipGraduate students and research mathematicians interested in number theory, in particular in Diophantine problems, quadratic forms, and lattices.

Table of Contents

Articles

Gabriele Nebe  Boris Venkov’s Theory of Lattices and Spherical Designs

Juan M. Cerviño and Georg Hein  Generalized Theta Series and Spherical Designs

Wai Kiu Chan and ByeongKweon Oh  Representations of integral quadratic polynomials

Renaud Coulangeon and Gabriele Nebe  Dense lattices as Hermitian tensor products

Rainer Dietmann  Small zeros of homogeneous cubic congruences

A. G. Earnest and Ji Young Kim  Strictly Regular Diagonal Positive Definite Quaternary Integral Quadratic Forms

Lenny Fukshansky  Heights and quadratic forms: Cassels’ theorem and its generalizations

Juan José Alba González and Florian Luca  On the positive integers $n$ satisfying the equation $F_n = x^2 + n y^2$

Jonathan Hanke  Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

D. R. HeathBrown  $p$adic Zeros of Systems of Quadratic Forms

David Kettlestrings and Jeffrey Lin Thunder  The Number of Function Fields with Given Genus

Gregory T. Minton  Unique Factorization in the Theory of Quadratic Forms

Gabriele Nebe  Golden lattices

Rudolf Scharlau  The extremal lattice of dimension 14, level 7 and its genus

Achill Schürmann  Strict Periodic Extreme Lattices

C. L. Stewart  Exceptional units and cyclic resultants, II

Jeffrey D. Vaaler and Martin Widmer  A note on generators of number fields

Takao Watanabe, Syouji Yano and Takuma Hayashi  Voronoï’s reduction theory of $GL_n$ over a totally real number field

Mark Watkins  Some comments about Indefinite LLL


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This volume contains the proceedings of the International Workshop on Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, held November 13–18, 2011, at the Banff International Research Station, Banff, Alberta, Canada.
The articles in this volume cover the arithmetic theory of quadratic forms and lattices, as well as the effective Diophantine analysis with height functions. Diophantine methods with the use of heights are usually based on geometry of numbers and ideas from lattice theory. The target of these methods often lies in the realm of quadratic forms theory. There are a variety of prominent research directions that lie at the intersection of these areas, a few of them presented in this volume:
 Representation problems for quadratic forms and lattices over global fields and rings, including counting representations of bounded height.
 Small zeros (with respect to height) of individual linear, quadratic, and cubic forms, originating in the work of Cassels and Siegel, and related Diophantine problems with the use of heights.
 Hermite's constant, geometry of numbers, explicit reduction theory of definite and indefinite quadratic forms, and various generalizations.
 Extremal lattice theory and spherical designs.
Graduate students and research mathematicians interested in number theory, in particular in Diophantine problems, quadratic forms, and lattices.

Articles

Gabriele Nebe  Boris Venkov’s Theory of Lattices and Spherical Designs

Juan M. Cerviño and Georg Hein  Generalized Theta Series and Spherical Designs

Wai Kiu Chan and ByeongKweon Oh  Representations of integral quadratic polynomials

Renaud Coulangeon and Gabriele Nebe  Dense lattices as Hermitian tensor products

Rainer Dietmann  Small zeros of homogeneous cubic congruences

A. G. Earnest and Ji Young Kim  Strictly Regular Diagonal Positive Definite Quaternary Integral Quadratic Forms

Lenny Fukshansky  Heights and quadratic forms: Cassels’ theorem and its generalizations

Juan José Alba González and Florian Luca  On the positive integers $n$ satisfying the equation $F_n = x^2 + n y^2$

Jonathan Hanke  Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

D. R. HeathBrown  $p$adic Zeros of Systems of Quadratic Forms

David Kettlestrings and Jeffrey Lin Thunder  The Number of Function Fields with Given Genus

Gregory T. Minton  Unique Factorization in the Theory of Quadratic Forms

Gabriele Nebe  Golden lattices

Rudolf Scharlau  The extremal lattice of dimension 14, level 7 and its genus

Achill Schürmann  Strict Periodic Extreme Lattices

C. L. Stewart  Exceptional units and cyclic resultants, II

Jeffrey D. Vaaler and Martin Widmer  A note on generators of number fields

Takao Watanabe, Syouji Yano and Takuma Hayashi  Voronoï’s reduction theory of $GL_n$ over a totally real number field

Mark Watkins  Some comments about Indefinite LLL