×

Large rational torsion on abelian varieties. (English) Zbl 0757.14025

From the introduction: The search for large rational torsion on elliptic curves goes back to the early 1900s, when Levi, Billing and Mahler, and others found, for various values of \(N\), the curve \(X_ 1(N)\) the rational points of which correspond to elliptic curves with rational \(N\)-torsion. These authors found rational points of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 12 on elliptic curves over \(\mathbb Q\), and proved the non-existence of various torsions. It follows from B. Mazur’s result in [Modular Funct. one Var. V, Proc. int. Conf., Bonn 1976, Lect. Notes Math. 601, 107-148 (1977; Zbl 0357.14005)] that no others may occur.
The search for large rational torsion on abelian varieties in theory entails finding rational points on associated higher dimensional varieties which are analogous to the \(X_ 1(N)\). In practice, it soon become apparent that the equations defining these varieties are far too large to be dealt with directly; we shall describe a technique which exploits additions of a certain simple type to obtain manageable subvarieties, on which rational points may more easily be sought. This method will be illustrated in section 3, in which new torsions over \(\mathbb Q\) are found.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14K99 Abelian varieties and schemes
14G05 Rational points

Citations:

Zbl 0357.14005
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billing, J.; Mahler, K., On exceptional points on cubic curves, J. London Math. Soc., 15, 32-43 (1940) · JFM 66.1211.01
[2] Cantor, D. G., Computing in the Jacobian of a hyperelliptic curve, Math. Comp., 48, 95-101 (1987) · Zbl 0613.14022
[3] Cassels, J. W.S, The Mordell-Weil group of curves of genus 2, (Arithmetic and Geometry Papers Dedicated to I.R. Shafarevich on the Occasion of this Sixtieth Birthday. Vol. 1. Arithmetic (1983), Birkhäuser: Birkhäuser Boston), 29-60 · Zbl 0529.14015
[4] Davenport, J. H., On the Integration of Algebraic Functions, (Lecture Notes in Computer Science, Vol. 102 (1981), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0661.26006
[5] D. GrantJ. Reine Angew. Math.; D. GrantJ. Reine Angew. Math. · Zbl 0702.14025
[6] Levi, B., Saggio per una teoria arithmetica delle forme cubiche ternaire, Atti R. Accad., 41, 739-764 (1906) · JFM 37.0251.01
[7] Mazur, B., Rational points of modular curves, (Modular Functions of One Variable, \(V\). Modular Functions of One Variable, \(V\), Lecture Notes in Mathematics, Vol. 601 (1977), Springer-Verlag: Springer-Verlag Berlin/New York), 107-148 · Zbl 0357.14005
[8] Reichert, M. A., Explicit determination of non-trivial torsion structures of elliptic curves over quadratic number fields, Math. Comp., 47, 637-658 (1986) · Zbl 0605.14028
[9] Silverman, J. H., (The Arithmetic of Elliptic Curves (1986), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0585.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.