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 The paper contains a hitherto unpublished result by M. Raynaud saying that if \(m\) is odd, then \(c_E\) is not divisible by 4.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

[1] Atkin, A.O.L. and Lehner, J. : Hecke operators on \Gamma 0(m) , Math. Ann. 185 (1970), 134-160. · Zbl 0177.34901 · doi:10.1007/BF01359701
[2] Bosch, S. , Lütkebohmert, W. , and Raynaud, M. : Néron Models , Springer-Verlag, New York, 1990. · Zbl 0705.14001
[3] Carayol, H. : Sur les représentations l-adiques associées aux formes modulaires de Hilbert , Ann. Sc. Ens 19 (1986), 409-468. · Zbl 0616.10025 · doi:10.24033/asens.1512
[4] Deligne, P. , and Rapoport, M. : Schémas de modules des courbes elliptiques, dans Modular functions of one variable II , Lectures Notes in Mathematics 349, Springer-Verlag, New York, 1973. · Zbl 0281.14010
[5] Edixhoven, B. : On the Manin constants of modular elliptic curves , in: Arithmetic algebraic geometry , Progress in Math. 89, pp. 25-39, Birkhäuser, 1989. · Zbl 0749.14025
[6] Mazur, B. : Modular curves and the Eisenstein ideal , Pub. Math. IHES 47 (1977), 33-186. · Zbl 0394.14008 · doi:10.1007/BF02684339
[7] Mazur, B. : Courbes Elliptiques et Symboles Modulaires , Séminaire Bourbaki, exposé 414, Juin 1972. · Zbl 0276.14012
[8] Mazur, B. : Rational isogenies of prime degree , Invent. Math. 44 (1978), 129-162. · Zbl 0386.14009 · doi:10.1007/BF01390348
[9] Raynaud, M. : Schémas en groupes de type (p, ... , p) , Bul. S.M. F. t. 102 (1974), 241-280. · Zbl 0325.14020 · doi:10.24033/bsmf.1779
[10] Ribet, K. : Endomorphisms of semi-stable abelian varieties over number fields , Ann. of Math. 101 (1975), 555-562. · Zbl 0305.14016 · doi:10.2307/1970941
[11] Shimura, G. : Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten , Publishers and Princeton University Press, 1971. · Zbl 0221.10029
[12] Tate, J. : p-Divisible groups, Proceedings of a conference on local fields, Nuffic summer school Driebergen, 1966, pp. 158-183. · Zbl 0157.27601
[13] Zagier, D. : Modular parametrizations of elliptic curves , Canad. Math. Bull. 28 (3) (1985), 372-384. · Zbl 0579.14027 · doi:10.4153/CMB-1985-044-8
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