From p-adic integration to motivic integration
The integration on p-adic number fields and the ring of the adeles of a number field was introduced by Tate to prove the functional equation of Hecke’s L-function. Tamagawa and Weil observed that the same technique can be used on the adelic points of a smooth algebraic variety over a number field, and that there is a closed relationship between the adelic integration and the L-function of the algebraic variety. The subject takes its geometric form with the famous lecture in Orsay by Kontsevich in 1995 where he proved that birationally equivalent complex Calabi–Yau varieties have the same Hodge numbers. That is the starting of the motivic integration. In this course, we will follow the historical line, explaining the basic ideas and results along the line.