Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T07:30:25.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Duality on tori and multiplicative dependence relations

Part of: Abelian varieties and schemes

Published online by Cambridge University Press:  09 April 2009

Daniel Bertrand
Daniel Bertrand
Affiliation:
Université de Paris VI Institut de Mathématiques, T. 46, C. 247 4, Place Jussieu 75 252 Paris Cédex 05France e-mail: bertrand@math.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give multihomogeneous estimates for the group of relations linking multiplicatively dependent algebraic numbers. In the process, we raise a question in the style of Lehmer's problem, concerning multidimensional covolumes in the lattice of units. The proofs are based on the Brill-Gordan duality theorem on orthogonal lattices, and the paper closes with an algebraic version of the theorem, concerning orthogonal abelian subvarieties of an arbitrarily polarized abelian variety.

Keywords

geometry of numbersregulatorsabelian varieties.

MSC classification

Secondary: 14K05: Algebraic theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 2 , April 1997 , pp. 198 - 216
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bertrand, D., ‘Minimal heights and polarizations on group varieties’, Duke Math. J. 80 (1995), 223–250.CrossRefGoogle Scholar
[2]Bertrand, D. and Philippon, P., ‘Sous-groupes algébriques de groupes algébriques commutatifs’, Illinois J. Math. 32 (1988), 263–280.CrossRefGoogle Scholar
[3]Cassels, J. W. S., An introduction to the geometry of numbers (Springer, Berlin, 1959).CrossRefGoogle Scholar
[4]Chen, G.-L., ‘Relations de dépendance linéaire entre des logarithmes de nombres algébriques’, Ann. Fac. Sc. Toulouse, to apper.Google Scholar
[5]David, S., ‘Points de petite hauteur sur les courbes elliptiques’, J. Number Theory, to appear.Google Scholar
[6]Friedman, E., ‘Analytic formulae for the regulator of a number field’, Invent. Math. 98 (1989), 599–622.CrossRefGoogle Scholar
[7]Hartshorne, R., Algebraic geometry (Springer, Berlin, 1977).CrossRefGoogle Scholar
[8]Heath-Brown, R., ‘Diophantine approximation with square-free numbers’, Math. Z. 187 (1984), 335–344.CrossRefGoogle Scholar
[9]Lange, H. and Birkenhake, C., Complex abelian varieties (Springer, Berlin, 1992).CrossRefGoogle Scholar
[10]Loxton, J. and van der Poorten, A., ‘Multiplicative dependence in number fields’, Acta Arith. 42 (1983), 291–302.CrossRefGoogle Scholar
[11]Masser, D. and Wüstholz, G., ‘Periods and minimal abelian subvarieties’, Ann. of Math. (2) 137 (1993), 407–458.CrossRefGoogle Scholar
[12]Matveev, E. M., ‘On linear and multiplicative relations’, Russian Ac. Sci. Sbornik Math. 78 (1994), 411–425.CrossRefGoogle Scholar
[13]Meyer, M. and Pajor's, A., ‘Volume des sections de la boule unité de λpn’, J. Funct. Anal. 80 (1988), 109–123.CrossRefGoogle Scholar
[14]Mumford, D., Abelian varieties (Oxford University Press, Oxford, 1970).Google Scholar
[15]Schatz, S., ‘Group schemes, formal groups and p-divisible groups’, in: Arithmetic geometry (ed. Cornell-Silverman, ) (Springer, Berlin, 1986).Google Scholar
[16]Serre, J-P., Groupes algébriques et corps de classes (Hermann, Paris, 1959).Google Scholar
[17]Thunder, J., ‘Asymptotic estimates for rational points of bounded height on flag varieties’, Compositio Math. 88 (1993), 155–186.Google Scholar