Inequality of the of the discriminant for elliptic pencils at any reductions

In this article the degree of the discriminant of an elliptic pencil on a p rojective curve is upper-bounded by using the degree of its conductor and t he genus of the base curve. This is done in the most general case, extendin g a method and a result of Szpiro (1981 and 1990a) and a result of Hindry a nd Silvermann. The difficult part, dealing with characteristic 2 and 3 and additive reductions, need the introduction of a new object - which we calle d '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 potentia lly good reduction cases; (2) the passage from the semi-stable reduction to the general case for a strong inequality.