Séminaire BOURBAKI Janvier 2012 64ème année, 2011-2012, n ...

1049–06
unconditionally. In the case of F q (t), the inequality r ≤ r an is known, so one deduces
Average(r) ≤ 2.3 in this setting. Over Q, the constant 2.3 was improved to 2 by Heath-
Brown [HB04, Theorem 1] and to 25/14 by M. Young [You06]. The latter implied the
(conditional) positivity of Prob(r an ≤ 1), and then also of Prob(r = r an ≤ 1), because
r an ≤ 1 implies r = r an ([Kol88, Kol90, GZ86] with [BFH90] or [MM91]).
Conditional bounds on Average(r an ) for other algebraic families of elliptic curves
and abelian varieties have been given by É. Fouvry and J. Pomykała [FP93],
P. Michel [Mic95, Mic97], J. Silverman [Sil98], and R. Wazir [Waz04].
2.2. Distribution of Selmer groups
For the family of elliptic curves y 2 =x 3 +k over Q, É. Fouvry proved Average(3 r/2 )