The propreties of an L-function from a geometric point of view

Abstract We will discuss similarities of L-functions between in the number theory and in the geometry. In particular the Hesse-Weil congruent zeta function of a smooth curve defined over a finite field will be compared to the zeta function associated to a disrete dynamical system. Moreover a geometric analog of the Birch and Swinnerton-Dyre conjecture and of the Iwasawa Main Conjecture will be discussed