Abstract
The height function on an abelian variety is addressed via an analogue of Mahler's measure function. It has previously been shown that the measure of a suitable polynomial yields the canonical height function on an elliptic curve; such work is generalised here to demonstrate that the canonical height on a higher-dimensional abelian variety may also be pursued from this viewpoint. Effective formulae which make use of the group law are given for the computation of local measures, and it is shown how these give rise to the construction of Riemann-style integrals on an abelian variety.
Preview
Unable to display preview. Download preview PDF.
References
Bertrand, D.: La Theorie de Baker Revisitée. Publ. Math. de l'Université de s VI, Problèmes Diophantiens exp. no.2 (1984/5)
Bourbaki, N.: Elements of Mathematics: Lie Groups and Lie Algebras. Hermann, Paris (1975)
Boyd, David W.: Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24 (1981) 453–469
Call, Gregory S., Silverman, Joseph H.: Canonical heights on varieties with morphisms. Compositio Mathematica 89 (1993) 163–205
Coates, John, Lang, Serge: Diophantine approximation on abelian varieties with complex multiplication. Inventiones Mathematicae 34 (1976) 129–133
Denef, J., van den Dries, L.: p-adic and real subanalytic sets. Annals of Mathematics 128 (1988) 79–138
Everest, G.R.: On the proximity of algebraic units to divisors. J. Number Theory (to appear)
Everest, G.R., ní Fhlathúin, Bríd: The elliptic Mahler measure. Math. Proc. Camb. Phil. Soc. 120 (1996) 13–25
ní Fhlathúin, Bríd: Mahler's Measure on an Abelian Variety, Ph.D. thesis. The University of East Anglia (1995)
Flynn, Eugene Victor: The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field. Math. Proc. Camb. Phil. Soc. 107 (1990) 425–441
Grant, D.: Formal groups in genus two. J. reine angew. Math. 411 (1990) 96–121
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)
Lang, S.: Fundamentals of Diophantine Geometry. Springer-Verlag (1983)
Lang, S.: Number Theory III: Diophantine Geometry. Springer-Verlag (1991)
Mahler, K.: An application of Jensen's formula to polynomials. Mathematika 7 (1960) 98–100
Mahler, K.: On some inequalities for polynomials in several variables. J. London Math. Soc. 37 (1962) 341–344
Néron, A.: Quasi-fonctions et hauteurs sur les variétés abéliennes. Annals of Mathematics 82 no. 2 (1965) 249–331
Silverman, Joseph H.: Advanced topics in the arithmetic of elliptic curves. Springer-Verlag (1994)
Silverman, Joseph H.: Computing heights on elliptic curves. Mathematics of Computation 51 (1988) 339–358
Serre, J-P.: Lectures on the Mordell-Weil Theorem. Vieweg, Braunschweig (1990)
Smyth, C.J.: On measures of polynomials in several variables. Bull. Austral. Math. Soc. 23 (1981) 49–63
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fhlathúin, B.n. (1996). The height on an abelian variety. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61581-4_47
Download citation
DOI: https://doi.org/10.1007/3-540-61581-4_47
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61581-1
Online ISBN: 978-3-540-70632-8
eBook Packages: Springer Book Archive