Conjectures in Arithmetic Algebraic Geometry

In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math­ ematicians for a long period of time. Starting from Fermat''s Last Theorem one is naturally led to introduce L­ functions, the main, motivation being the calculation of class numbers. In partic­ ular, Kummer showed that the class numbers of cycl...