Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2025-01-03T00:18:44.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Canonical heights and division polynomials

Published online by Cambridge University Press:  14 August 2014

ROBIN de JONG  and
J. STEFFEN MÜLLER
ROBIN de JONG
Affiliation:
Mathematisch Instituut, Universiteit Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands. e-mail: [email protected]
J. STEFFEN MÜLLER
Affiliation:
Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss a new method to compute the canonical height of an algebraic point on a hyperelliptic jacobian over a number field. The method does not require any geometrical models, neither p-adic nor complex analytic ones. In the case of genus 2 we also present a version that requires no factorisation at all. The method is based on a recurrence relation for the ‘division polynomials’ associated to hyperelliptic jacobians, and a diophantine approximation result due to Faltings.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 357 - 373
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Cassels, J. W. S. and Flynn, E. V.. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series no. 230 (Cambridge University Press, 1996).CrossRefGoogle Scholar
[2]David, S. and Hirata–Kohno, N.. Linear forms in elliptic logarithms. J. Reine Angew. Math. 628 (2009), 37–89.Google Scholar
[3]Everest, G. and Ward, T.. The canonical height of an algebraic point on an elliptic curve. New York J. Math. 6 (2000), 331–342.Google Scholar
[4]Faltings, G.. Diophantine approximation on abelian varieties. Ann. of Math. (2) 133 (1991), 549–576.CrossRefGoogle Scholar
[5]Flynn, E. V. and Smart, N. P.. Canonical heights on the Jacobians of curves of genus 2 and the infinite descent. Acta Arith. 79 (1997), 333–352.CrossRefGoogle Scholar
[6]Holmes, D.. Computing Néron–Tate heights of points on hyperelliptic Jacobians. J. Number Theory 132 (2012), 1295–1305.CrossRefGoogle Scholar
[7]Kanayama, N.. Division polynomials and multiplication formulae of Jacobian varieties of dimension 2. Math. Proc. Camb. Phil. Soc. 139 (2005), 399–409.CrossRefGoogle Scholar
[8]Kanayama, N.. Corrections to “Division polynomials and multiplication formulae in dimension 2”. Math. Proc. Camb. Phil. Soc. 149 (2010), 189–192.CrossRefGoogle Scholar
[9]Lang, S.. Higher dimensional diophantine problems. Bull. Amer. Math. Soc. 80 (1974), 779–787.CrossRefGoogle Scholar
[10]Lang, S.. Fundamentals of Diophantine Geometry (Springer–Verlag, 1983).CrossRefGoogle Scholar
[11]MAGMA is described in Bosma, W., Cannon, J. and Playoust, C.. The Magma algebra system I: The user language. J. Symb. Comp. 24 (1997), 235–265.CrossRefGoogle Scholar
[12]Müller, J. S.. Computing canonical heights using arithmetic intersection theory. Math. Comp. 83 (2014), 311–336.CrossRefGoogle Scholar
[13]Müller, J. S.. Explicit Kummer varieties of hyperelliptic Jacobian threefolds, to appear in LMS J. Comput. Math. (2014).CrossRefGoogle Scholar
[14]Müller, J. S. and Stoll, M.. Canonical heights on genus two Jacobians. In preparation.Google Scholar
[15]Stoll, M.. On the height constant for curves of genus two, II. Acta Arith. 104 (2002), 165–182.CrossRefGoogle Scholar
[16]Stoll, M.. An explicit theory of heights for hyperelliptic Jacobians of genus three. In preparation.Google Scholar
[17]Uchida, Y.. Canonical local heights and multiplication formulas. Acta Arith. 149 (2011), 111–130.CrossRefGoogle Scholar
[18]Uchida, Y.. Division polynomials and canonical local heights on hyperelliptic Jacobians. Manuscript. Math. 134 (2011), 273–308.CrossRefGoogle Scholar
[19]Uchida, Y.. Valuations of Somos 4 sequences and canonical local heights on elliptic curves. Math. Proc. Camb. Phil. Soc. 150 (2011), 385–397.CrossRefGoogle Scholar