  On the rank of Jacobians over function fields.} Let $f:\mathcal{X}\to C$ be a
projective surface fibered over a curve and defined over a number field $k$. We
give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the
jacobian of the generic fibre (modulo the constant part) in terms of average of
the traces of Frobenius on the fibers of $f$. The results also give a
reinterpretation of the Tate conjecture for the surface $\mathcal{X}$ and
generalizes results of Nagao, Rosen-Silverman and Wazir.
