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An Arithmetical Function Associated With the Rank of Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

David Clark
David Clark*
Affiliation:
Department of Mathematics and Statistics McGill University Montréal, Quebec, H3A 2K6
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Abstract

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We define an arithmetical function, f(n), which gives a lower bound for the rank of elliptic curves, y2 = x3 + nx, n square-free. Thus, if f(n) is unbounded for square-free values of n, then there are elliptic curves of arbitrarily large rank. We show that f(n) is unbounded as n ranges over all integers.

Keywords

11G0511D85.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 2 , 01 June 1991 , pp. 181 - 185
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Clark, D., “L-series and Ranks of Elliptic Curves,” M. Sc. Thesis, McGill University, Montréal, Québec, 1988.Google Scholar
2. Coates, J., Elliptic Curves and Iwasawa Theory, in “Modular Forms,” R. Rankin éd., Halsted Press, New York, 1984.Google Scholar
3. Lutz, E., Sur Vequation y2 = x3 — Ax—B dans le corps p-adic, J. Reine Angew. Math. 177 (1937), 237–247.Google Scholar
4. Mazur, B., Modular curves and the Eisenstein ideal, IHES Publ. Math. 47 (1977), 33–186.Google Scholar
5. Mestre, J.-F., Formules explicites et minorations de conducteurs de variétés algébriques, Comp. Math. 58 (1986), 209–232.Google Scholar
6. Mordell, L. J., On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Camb. Phil. Soc. 21 (1922), 179–192.Google Scholar
7. Nagell, T., Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I (1935), Nr. 1.Google Scholar
8. Néron, A., Problèmes arithemétiques et géométriques rattachés a la notion de rang d'une courbe algégbrique dans un corps, Bull. Soc. Math. France 80 (1952), 101–166.Google Scholar
9. Shafarevich, I. R. and Tate, J., The rank of elliptic curves, A.M.S. Transi. 8 (1967), 917–920.Google Scholar