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MULTIPLICATIVE SUBGROUPS OF J0(N) AND APPLICATIONS TO ELLIPTIC CURVES

Published online by Cambridge University Press:  08 March 2005

V. Vatsal
V. Vatsal
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada (vatsal@math.ubc.ca)
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Abstract

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In this paper, we prove a certain maximality property of the Shimura subgroup amongst the multiplicative-type subgroups of $J_0(N)$, and apply this to verify conjectures of Stevens on the existence of certain canonical parametrizations of rational elliptic curves by modular curves. We are also able to verify some of Stevens’s conjectures on the characterization of the elliptic curve in an isogeny class with minimal Faltings–Parshin height.

AMS 2000 Mathematics subject classification: Primary 11G05; 11G18

Keywords

elliptic curvesmodular curvescomplex multiplication
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 4 , Issue 2 , April 2005 , pp. 281 - 316
Copyright
2005 Cambridge University Press