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Modular Parametrizations of Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

D. Zagier
D. Zagier*
Affiliation:
University of MarylandCollege Park Maryland, U.S.A. Max-planck-institut für mathematik, BonnFRG
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Abstract

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Many — conjecturally all — elliptic curves E/ have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X0(N) to E such that the pull-back of a holomorphic differential on E is a modular form (newform) f of weight 2 and level N. We describe an algorithm for computing the degree of φ as a branched covering, discuss the relationship of this degree to the "congruence primes" for f (the primes modulo which there are congruences between f and other newforms), and give estimates for the size of this degree as a function of N.

Keywords

14K0710D1210D23
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 3 , 01 September 1985 , pp. 372 - 384
Copyright
Copyright © Canadian Mathematical Society 1985

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