The Weil conjectures, proven (in stages) by Weil, Dwork, Grothendieck and Deligne, give an asymptotic estimate on the number of solutions to a system of diophantine equations modulo p in the field of p^n elements, when n grows. These estimates stem from an archimedean estimate on the "eigenvalues of Frobenius" (the Riemann hypothesis over finite fields). We shall give two talks whose theme is what can be learned from p-adic estimates on these eigenvalues. The first talk will be elementary and will cover (1) a review of the Weil conjectures (2) elliptic curves - supersingular versus ordinary (3) Stickelberger's theorem on Gauss sums and an application to the Fermat curve. In the second talk (after the break) we shall talk about Newton and Hodge polygons, crystals and Katz's conjecture (proved by Mazur).