Abstract
During 1969 I was a guest lecturer in Japan, teaching a course in zeta functions and p-adic analysis at Kyoto University. These notes are essentially the lecture notes for that course. The first term, I presented several "classical" results on zeta functions in characteristic p : Weil's calculation of the zeta function of a diagonal hypersurface, Grothendieck's proof of the PRiemann hypothesisn for curves via the Riemann~Roch theorem for cnrf.Qf"'Pc:, and Dwork's proof of rationality. The second term was increasing p-adic. After sketching Serre's spectral theory for compact operators I gave a version of Dwork's first paper on the zeta function of a non-singular hypersurface, stressing the "Lefschetz fixed point theorem" character of the proof. Finally some indications of the connections between Dwork's differential operator theory and various cohomology theories, classical and otherwise were given, closely following Katz's thesis.