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A Mordell-Weil group of rank 8, and a subgroup of finite index

Published online by Cambridge University Press:  22 January 2016

Charles F. Schwartz
Charles F. Schwartz*
Affiliation:
Department of Mathematics, Rider College, P.O. Box 6400, Lawrenceville New Jersey 08640, USA
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It is well known [c.f. Kas] that every elliptic surface, with geometric genus 0, is given by a Weierstrass equation of the form

(relative to a suitable parameter, u, for the base) where the a’s and b’s are constants. For sufficiently general choices of a’s and b’s, the Mordell-Weil group (i.e., the group of solutions (x, y), with x and y rational functions of u) has rank 8.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 93 , March 1984 , pp. 19 - 26
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Cox, D. A. and Zucker, S., Intersection numbers of sections of elliptic surfaces, Invent. Math., 53 (1979), 1–44.CrossRefGoogle Scholar
[ 2 ] Schwartz, C. F., On a family of elliptic surfaces with Mordell-Weil rank 4, to appear.Google Scholar
[ 3 ] Shioda, T., On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20–59.CrossRefGoogle Scholar
[ 4 ] Schwartz, C. F., An elliptic surface of Mordell-Weil rank 8 over the rational numbers, to appear.Google Scholar