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Modular parametrizations of elliptic curves. (English) Zbl 0579.14027
A famous conjecture of Taniyama and Weil states the existence of a modular parametrization $$\phi$$ : $$X_ 0(N)\to E$$ for every elliptic curve E defined over $${\mathbb{Q}}$$. Here $$X_ 0(N)$$ denotes the quotient of the complex upper halfplane by the Hecke congruence subgroup $$\Gamma_ 0(N)$$ of $$SL_ 2({\mathbb{Z}}).$$
In the article under consideration, the degree of $$\phi$$ is investigated, the existence taken for granted. Let f be a normalized new form for $$\Gamma_ 0(N)$$, corresponding to $$\phi$$ as above. The relation $4\pi^ 2 \| f\|^ 2=\deg (\phi) vol(E)$ reduces the computation of deg($$\phi)$$ to that of the Petersson norm $$\| f\|$$ $$(vol({\mathbb{E}}))=covolume$$ of the lattice associated with E, being easy to compute numerically). In theorem 1, a formula for $$\| f\|^ 2$$ is given, permitting the determination of deg($$\phi)$$ by means of the Fourier coefficients of f (corollary of theorem 2). For all strong Weil curves of prime conductor $$N<200$$, deg($$\phi)$$ is tabulated, complementing the tables in the book ”Modular functions of one variable. IV”, edited by B. J. Birch and W. Kuyk [Lect. Notes math. 476 (1975; Zbl 0315.14014)]. In theorem 3, the author proves $$(N=prime$$, $$\phi =strong$$ Weil parametrization): $$\deg (\phi)=l\arg est$$ number r such that $$f\equiv g (mod r)$$ for a suitable g with integer Fourier coefficients, weight 2 and $$g\perp f.$$
Finally, conditional upper and lower bounds (depending on the Taniyama- Weil and Manin conjectures) for deg($$\phi)$$ and varying N are discussed.
Reviewer: E.Gekeler

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14H52 Elliptic curves 11F33 Congruences for modular and $$p$$-adic modular forms 11F11 Holomorphic modular forms of integral weight
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