properties of the eigenvalues of Frobenius via examples. In the second talk, we shall explain what crystalline cohomology is. It is in fact very close to the "familiar" de Rham cohomology based on
differential forms. We shall then talk about Hodge numbers and what they can tell us about the
p-adic absolute values of the eigenvalues of Frobenius. We shall present Katz' conjecture (proved by Mazur) that the Newton Polygon lies above the Hodge polygon. We shall describe without proofs some results about how these notions behave in families, especially families coming from moduli spaces of abelian varieties.