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Modular Parametrizations of Elliptic Curves
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- Journal:
- Canadian Mathematical Bulletin / Volume 28 / Issue 3 / 01 September 1985
- Published online by Cambridge University Press:
- 20 November 2018, pp. 372-384
- Print publication:
- 01 September 1985
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- Article
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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.
have a "modular parametrization," i.e. for some 