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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 The paper contains a hitherto unpublished result by M. Raynaud saying that if \(m\) is odd, then \(c_E\) is not divisible by 4.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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