Lattice point counting and height bounds over number fields and quaternion algebras
| Autor: | Fukshansky, Lenny, Henshaw, Glenn |
|---|---|
| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Online Journal of Analytic Combinatorics, vol. 8 (2013), art. 4, 20 pp |
| Druh dokumentu: | Working Paper |
| Popis: | An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a collection of height comparison inequalities for heights over a number field and over a quaternion algebra. We also show how these inequalities can be used to obtain existence results for points of bounded height over a quaternion algebra, which constitute non-commutative analogues of variations of the classical Siegel's lemma and Cassels' theorem on small zeros of quadratic forms. Comment: 22 pages; to appear in the Online Journal of Analytic Combinatorics |
| Databáze: | arXiv |
| Externí odkaz: |
|