Modular Parametrizations of Elliptic Curves
D. Zagier
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Source: Crossref
Published: Sep 1, 1985
DOI: 10.4153/cmb-1985-044-8
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Abstract Many — conjecturally all — elliptic curves E / have a "modular parametrization," i.e. for some N there is a map φ from the modular curve X 0 ( 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 .
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