Lattice point counting and height bounds over number fields and quaternion algebras

Lenny Fukshansky, Glenn Henshaw

Abstract


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.

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References


F. Barroero. Algebraic integers of fixed degree and bounded height. preprint; arXiv:1305.0482v2.

Y. Bugeaud. Bounds for the solutions of superelliptic equations. Compositio Math., 107(2):187–219, 1997.

Y. Bugeaud and K. Gyory. Bounds for the solutions of unit equations. Acta Arith., 74(1):67– 80, 1996.

J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer, Berlin, 1959.

W. K. Chan and L. Fukshansky. Small zeros of hermitian forms over quaternion algebras. Acta Arith., 142(3):251–266, 2010.

W. K. Chan, L. Fukshansky, and G. Henshaw. Small zeros of quadratic forms missing a union of varieties. preprint, http://math.cmc.edu/lenny/papers/quad_zero-1.pdf, 2012.

R. Dietmann. Small zeros of quadratic forms avoiding a finite number of prescribed hyperplanes. Canad. Math. Bull., 52(1):63–65, 2009.

G. R. Everest and J. H. Loxton. Counting algebraic units with bounded height. J. Number Theory, 44(2):222–227, 1993.

C. Fuchs, R. Tichy, and V. Ziegler. On quantitative aspects of the unit sum number problem. Arch. Math. (Basel), 93(3):259–268, 2009.

L. Fukshansky. Small zeros of quadratic forms with linear conditions. J. Number Theory, 108(1):29–43, 2004.

L. Fukshansky. Integral points of small height outside of a hypersurface. Monatshefte fu ̈r Mathematik, 147(1):25–41, 2006.

L. Fukshansky. Siegel’s lemma with additional conditions. J. Number Theory, 120(1):13–25, 2006.

L. Fukshansky. Algebraic points of small height missing a union of varieties. J. Number Theory, 130(10):2099–2118, 2010.

E. Gaudron. Geometrie des nombres ad ́elique et lemmes de Siegel g ́en ́eralis ́es. Manuscripta Math., 130(2):159182, 2009.

P. Gritzmann and J. M. Wills. Lattice points. In Handbook of Convex Geometry, Vol. A, B, pages 765–797. North-Holland, Amsterdam, 1993.

S. Lang. Fundamentals of Diophantine geometry. Springer-Verlag, 1983.

S. Lang. Algebraic Number Theory. Springer-Verlag, 1994.

C. Liebendorfer. Linear equations and heights over division algebras. J. Number Theory, 105(1):101–133, 2004.

C. Liebendorfer. Heights and determinants over quaternion algebras. Comm. Algebra, 33(10):3699–3717, 2005.

C. Liebendorfer and G. R ́emond. Duality of heights over quaternion algebras. Monatsh. Math., 145(1):61–72, 2005.

T. Loher and D. Masser. Uniformly counting points of bounded height. Acta Arith., 111(3):277–297, 2004.

D. Masser and J. D. Vaaler. Counting algebraic numbers with large height. II. Trans. Amer. Math. Soc., 359(1):427–445, 2007.

D. G. Northcott. An inequality in the theory of arithmetic on algebraic varieties. Proc. Camb. Phil. Soc., 45:502–509 and 510–518, 1949.

R. S. Pierce. Associative Algebras. Springer-Verlag, 1982.

S. Schanuel. Heights in number fields. Bull. Soc. Math. France, 107(4):433–449, 1979.

W. M. Schmidt. Northcott’s theorem on heights. I. A general estimate. Monatsh. Math., 115(1-2):169–181, 1993.

M. A. Tsfasman and S. G. Vladut. Algebraic-Geometric Codes. Kluwer Academic Publishers, 1991.

J. D. Vaaler. Small zeros of quadratic forms over number fields. Trans. Amer. Math. Soc., 302(1):281–296, 1987.

J. D. Vaaler. Small zeros of quadratic forms over number fields, II. Trans. Amer. Math. Soc., 313(2):671–686, 1989.

T. Watanabe. Minkowski’s second theorem over a simple algebra. Monatsh. Math., 149(2):155–172, 2006.

M. Widmer. Integral points of fixed degree and bounded height. preprint.

M. Widmer. Counting points of fixed degree and bounded height. Acta Arith., 140(2):145–168, 2009.


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