We consider a family of abelian varieties over a number field K , i.e. a variety X with a map to a curve B whose fibres are abelian varieties (the interesting cases are when B is the projective line or an elliptic curve with positive rank). The generic fibre is an abelian variety over the function field K(B) and the group of K(B)-rational points has a rank r. For almost all points t in B(K) the fibre is an abelian variety Xt over K and the group of K-rational point has rank r(t). A specialisation theorem of Silverman says that for or almost all points t in B(K) the rank r(t) is greater or equal to r. We want to understand the distribution of r(t), in particular we ask wether there are infinitely many t's 1) with r(t)=r, 2) with r(t)>r. The problem looks very hard in general, but, under specific geometric conditions, we will settle the second question, and provide interesting example where much more can be proven. This is a joint work with Cecília Salgado.