Rang moyen de familles de courbes elliptiques et lois de Sato-Tate

The goal of this note is to give some complements to an article of Fouvery and Pomykala: by anad-hoc method, they bound on average the rank of elliptic curves over in polynomial families:y ²=x ³=a(t)x+b(t) whent varies in under some generic conditions on the polynomials (over a(t),b(t). Here, by a more systematic treatment, we are able to relax most of these conditions, keeping only the natural one (the family is not geometricaly trivial). However, this result, specialized to the case treated by Fouvry and Pomykala, yields a better bound; our method depends on the distribution of the number of points in families of elliptic curves over finite fields (known as the vertical Sato-Tate law), which itself depends on the work of Deligne on the Weil conjectures.