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On the Manin conjecture for modular elliptic curves. (A propos de la conjecture de Manin pour les courbes elliptiques modulaires.) (French) Zbl 0865.11049
Let $$E$$ be a strong Weil curve over $$\mathbb{Q}$$ with modular parametrization $$\varphi: X_0(N) \to E$$. Let $$\alpha_E$$ be a Néron differential on $$E$$. Then its pull-back satisfies $$\varphi^* (\alpha_E) =c_Ef_E dq/q$$, where $$f_E$$ is a normalized newform of weight 2 for $$\Gamma_0(N)$$ and $$c_E$$, the so-called Manin constant is a rational number, which can assumed to be positive by suitably choosing the sign of $$\alpha_E$$. Manin conjectures that $$c_E=1$$.
Let $$m$$ be the largest number such that $$m^2$$ divides $$N$$. It is known that $$c_E$$ is an integer [B. Edixhoven in: Arithmetic algebraic geometry, Prog. Math. 89, 25-39 (1991; Zbl 0749.14025)] and that all prime divisors of $$c_E$$ divide $$2m$$ [B. Mazur, Invent. Math. 44, 129-162 (1978; Zbl 0386.14009)]. The paper contains a hitherto unpublished result by M. Raynaud saying that if $$m$$ is odd, then $$c_E$$ is not divisible by 4. This is based on a study of the failure of the Néron model functor to be exact when the ramification index $$e$$ equals $$p-1$$. The relevant results are given in an appendix.
On the other hand, the authors prove that all prime divisors of $$c_E$$ must divide $$N$$, which provides new information when $$N$$ is odd. As a corollary, they obtain Manin’s conjecture for semi-stable elliptic curves with good reduction at 2. They further remark that specific results of Raynaud’s imply the truth of the Manin conjecture for all other semi-stable strong Weil curves $$E$$, except when $$E$$ has multiplicative reduction at 2 and the minimal discriminant of $$E$$ has even 2-adic valuation.

##### MSC:
 11G05 Elliptic curves over global fields 14H52 Elliptic curves 14G35 Modular and Shimura varieties
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##### References:
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