Rang des courbes elliptiques et sommes d'exponentielles

Abstract

Lety2=x3+a(t)x+b(t) be the equation of an elliptic curve over ℚ, wherea(t) andb(t) are polynomials overℤ. We give an upper bound on average of the rank of such curves whent varies overℤ under the assumption that theL-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Deligne's theorem.