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 generalisations. Extremal lattice theory and spherical designs.
Boris Venkov's theory of lattices and spherical designs by G. Nebe
Generalized theta series and spherical designs by J. M. Cervino and G. Hein
Representations of integral quadratic polynomials by W. K. Chan and B.-K. Oh
Dense lattices as Hermitian tensor products by R. Coulangeon and G. Nebe
Small zeros of homogeneous cubic congruences by R. Dietmann Strictly regular
diagonal positive definite quaternary integral quadratic forms by A. G.
Earnest and J. Y. Kim Heights and quadratic forms: Cassels' theorem and its
generalizations by L. Fukshansky On the positive integers $n$ satisfying the
equation $F_n=x^2+ny^2$ by J. J. A. Gonzalez and F. Luca Algorithms for
computing maximal lattices in bilinear (and quadratic) spaces over number
fields by J. Hanke $p$ adic zeros of systems of quadratic forms by D. R.
Heath-Brown The number of function fields with given genus by D.
Kettlestrings and J. L. Thunder Unique factorization in the theory of
quadratic forms by G. T. Minton Golden lattices by G. Nebe The extremal
lattice of dimension 14, level 7 and its genus by R. Scharlau Strict periodic
extreme lattices by A. Schurmann Exceptional units and cyclic resultants, II
by C.L. Stewart A note on generators of number fields by J. D. Vaaler and M.
Widmer Voronoi's reduction theory of $GL_n$ over a totally real number field
by T. Watanabe, S. Yano, and T. Hayashi Some comments about indefinite LLL by
M. Watkins
Wai Kiu Chan, Wesleyan University, Middletown, CT, Lenny Fukshansky, Claremont McKenna College, CA, Rainer Schulze-Pillot, Universitat des Saarlandes, Saarbrucken, Germany, and Jeffrey D. Vaaler, University of Texas at Austin, TX, Editors
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