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Courbes de Weil semi-stables de discriminant une puissance m-ième. (Semistable Weil curves of m-th power discriminant). (French) Zbl 0693.14004

The authors examine semi-stable elliptic Weil curves E defined over \({\mathbb{Q}}\), whose minimal discriminants \(\Delta\) are m-th powers. (“Weil curve”: There exists a non-constant \({\mathbb{Q}}\)-morphism of some modular curve \(X_ 0(N)\) to E.) The main result is theorem 1:
If E is as above and such that \(| \Delta |\) is an m-th power, then \(m\leq 5\), and E has an m-division point over \({\mathbb{Q}}.\)
As a consequence, the authors also show theorem 2:
If the conductor of E is a prime p, then (with few exceptions explicitly described) the minimal discriminant \(\Delta\) equals \(\pm p.\)
In the proof of theorem 1, prime divisors \(\ell \geq 11\) of m are excluded by combining K. A. Ribet’s result [Invent. Math. 100, No.2, 431-476 (1990)] with an argument of J.-P. Serre [Duke Math. J. 54, 179-230 (1987; Zbl 0641.10026)]. In difficult case by case considerations, it is then shown that m cannot be divisible by 7, 10, 15, 6, 9, 25, 8, which gives the bound \(m\leq 5\).
Reviewer: E.-U.Gekeler

MSC:

14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
11F11 Holomorphic modular forms of integral weight
14G25 Global ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus

Citations:

Zbl 0641.10026
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