Arithmetic of elliptic curves

The problem on the arithmetic of elliptic curves is one of the oldest and still unsolved mathematical problems. The presupposed answer is an excellent example of modern theory.
This problem, that is due to Fermat, has been studied intensively during this century. During last thirty years these investigations were motivated by the famous Birch and Swinnerton-Dyer Conjecture that establish mysterious relation between the group of rational points on elliptic curve and L-functions. Moreover, this subject has some unexpected connections with such problems of number theory as Fermat Last Theorem and Gauss problem on the effective determination of all quadratic fields with a given class number.
All these aspects will be discussed in the lecture. The lecture is intended for the wide mathematical audience.