**Contemporary Mathematics**

Volume: 587;
2013;
243 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-0-8218-8318-1
Product Code: CONM/587**

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**Electronic ISBN: 978-0-8218-9503-0
Product Code: CONM/587.E**

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# Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms

Share this page *Edited by *
*Wai Kiu Chan; Lenny Fukshansky; Rainer Schulze-Pillot; Jeffrey D. Vaaler*

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.

#### Readership

Graduate students and research mathematicians interested in number theory, in particular in Diophantine problems, quadratic forms, and lattices.

# Table of Contents

## Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms

- In Memoriam v6 free
- Preface ix10 free
- Boris Venkov’s Theory of Lattices and Spherical Designs 114 free
- Generalized Theta Series and Spherical Designs 2134
- Representations of integral quadratic polynomials 3144
- Dense lattices as Hermitian tensor products 4760
- Small zeros of homogeneous cubic congruences 5972
- Strictly Regular Diagonal Positive Definite Quaternary Integral Quadratic Forms 6982
- Heights and quadratic forms: Cassels’ theorem and its generalizations 7790
- 1. Introduction: Cassels’ theorem 7790
- 2. Notation and heights 7992
- 3. Extensions over global fields 8295
- 4. Multiple zeros and isotropic subspaces 8396
- 5. Effective structure theorems 8598
- 6. Effective results with additional conditions 87100
- 7. Open problems 89102
- Acknowledgment 90103
- References 90103

- On the positive integers 𝑛 satisfying the equation 𝐹_{𝑛}=𝑥²+𝑛𝑦² 95108
- Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields 111124
- 𝑝-adic Zeros of Systems of Quadratic Forms 131144
- The Number of Function Fields with Given Genus 141154
- Unique Factorization in the Theory of Quadratic Forms 151164
- Golden lattices 157170
- The extremal lattice of dimension 14, level 7 and its genus 167180
- Strict Periodic Extreme Lattices 185198
- Exceptional units and cyclic resultants, II 191204
- A note on generators of number fields 201214
- Voronoï’s reduction theory of 𝐺𝐿_{𝑛} over a totally real number field 213226
- Some comments about Indefinite LLL 233246