Abstract
In this article the degree of the discriminant of an elliptic pencil on a projective curve is upper-bounded by using the degree of its conductor and the genus of the base curve. This is done in the most general case, extending a method and a result of Szpiro (1981 and 1990a) and a result of Hindry and Silvermann. The difficult part, dealing with characteristic 2 and 3 and additive reductions, need the introduction of a new object - which we called 'conducteur efficace' - defined by using differentials and interestingly comparable to the usual conductor. This article ends with a few results in the arithmetical case - case corresponding to an inequality conjectured by the second author in 1978: (1) the proof of this inequality in the potentially good reduction cases; (2) the passage from the semi-stable reduction to the general case for a strong inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Références
Flexor, M. and Oesterlé J.: Sur les points de torsion des courbes elliptiques, Astérique 183 (1990), 25-36.
Hartshorne, R.: Algebraic Geometry, Springer-Verlag, Berlin, 1977.
Hindry, M. and Silvermann, J. H.: The canonical height and integral point on elliptic curves, Invent. Math. 93 (1988), 419-450.
Liu, Q. and Saito, T.: Inequality for conductor and differentials of a curve over a local field, Prépublication de l'Université Bordeaux 1 (1998), 84.
Masser, D. W.: Note on a conjecture of Szpiro, Astérisque 183 (1990), 19-23.
Milne, J.: Étale Cohomology, Princeton Univ. Press, New Jersey, 1980.
Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. I.H.E.S. 21 (1964), 5-128.
Ogg, A. P.: Elliptic function with wild ramification, Amer. J. Math. 89 (1967), 1-21.
Pesenti, J.: Thèse de doctorat, Université d'Orsay (1998).
Raynaud, M.: Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes, Séminaire Bourbaki (1965), p. 286.
Saito, T: Conductor, discriminant and Noether formula of arithmetic surfaces, Duke Math. J. 57(1) (1988), 151-173.
Serre, J.-P.: Corps locaux, Hermann, Paris, 1968.
Silvermann, J. H.: Advanced Topics about Elliptic Curves, Springer-Verlag, Berlin, 1986.
Silvermann, J. H.: The Arithmetic of Elliptic Curves, Springer-Verlag, Berlin, 1986.
Szpiro L.: Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque 86 (1981).
Szpiro L.: Discriminant et conducteur, Astérisque 183 (1990), 7-18.
Szouri L.: Séminaire sur les pinceaux de courbes elliptiques, Astérisque 183 (1990).
Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil, In: Modular Functions in One Variable IV, Lectures Notes in Maths. 476, Springer-Verlag, 1975, pp. 33-52.
Voloch, F.: On the conjectures of Mordell and Lang in positive characteristic, Invent. Math. 104 (1991), 643-646.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pesenti, J., Szpiro, L. Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques. Compositio Mathematica 120, 83–117 (2000). https://doi.org/10.1023/A:1001736823128
Issue Date:
DOI: https://doi.org/10.1023/A:1001736823128