Modular parametrizations of elliptic curves. (English) Zbl 0579.14027

Zagier, Don

Can. Math. Bull. 28, 372-384 (1985).

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

Keywords:

degree of modular parametrization; congruence numbers; conjecture of Taniyama and Weil; elliptic curve; Petersson norm

Citations:

Zbl 0315.14014