Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques

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.