Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms
About this Title
Wai Kiu Chan, Wesleyan University, Middletown, CT, Lenny Fukshansky, Claremont McKenna College, Claremont, CA, Rainer Schulze-Pillot, Universität des Saarlandes, Saarbrucken, Germany and Jeffrey D. Vaaler, University of Texas at Austin, Austin, TX, Editors
Publication: Contemporary Mathematics
Publication Year 2013: Volume 587
ISBNs: 978-0-8218-8318-1 (print); 978-0-8218-9503-0 (online)
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.
Table of Contents
- 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 Byeong-Kweon 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. Heath-Brown – $p$-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