Let X be a smooth projective variety over a number field k, and consider the representation of the absolute Galois group of k on the second etale cohomology of X. An open conjecture of J.Tate says that the Hasse-Weil L-function associated to this representation has a pole at s=2 with order equal to the rank of the Neron- Severi group of X. In known cases when this conjecture holds, we can say a lot about the arithmetic of X.

In the same vein, for an elliptic fibration X->C defined over k, K.Nagao formulated a conjectural formula for the average of the p-coefficients of the L-series of the fibres. The conjecture predicts that this average is equal to the rank of the Mordell- Weil group X(k(C)). Rosen, Silverman and later Wazir proved that Tate´s conjecture implies Nagao´s formula. As an application, we get a simple proof that the distribution of p- coefficients of the L-series of two non isogenous elliptic curves over k are non- correlated.

This is the first of two talks. In this first talk I will define the concepts involved and sketch the ideas of some proofs. No prior knowledge of elliptic surfaces or Hasse-Weil L-functions is needed to understand the talk.