We give an overview of some questions and results in integral $p$-adic Hodge theory. A few proofs are supplied.

The purpose of the present manuscript is to continue the survey of the Hodge-Arakelov theory of elliptic curves (cf. [7], [8], [9], [10], [11]) that was begun in [12]. This theory is a sort of "Hodge theory of elliptic curves" analogous to the classical complex and $p$-adic Hodge theories, but which exists in the global arithmetic framework of Arakelov theory. In particular, in the present manuscript, we focus on the aspects of the theory (cf. [9], [10], [11]) developed subsequent to those discussed in [12], but prior to the conference "Algebraic Geometry 2000" held in Nagano, Japan, in July 2000. These developments center around the natural connection that exists on the pair consisting of the universal extension of an elliptic curve, equipped with an ample line bundle. This connection gives rise to a natural object — which we call the crystalline theta object — which exhibits many interesting and unexpected properties. These properties allow one, in particular, to understand at a rigorous mathematical level the (hitherto purely "philosophical") relationship between the classical Kodaira-Spencer morphism and the Galois-theoretic "arithmetic Kodaira-Spencer morphism" of Hodge-Arakelov theory. They also provide a method (under certain conditions) for "eliminating the Gaussian poles," which are the main obstruction to applying Hodge-Arakelov theory to diophantine geometry. Finally, these techniques allow one to give a new proof of the main result of [7] using characteristic $p$ methods. It is the hope of the author to survey more recent developments (i.e., developments that occurred subsequent to "Algebraic Geometry 2000") in a sequel to the present manuscript.

We explain the theory of refined cycle maps associated to arithmetic mixed sheaves. This includes the case of arithmetic mixed Hodge structures, and is closely related to work of Asakura, Beilinson, Bloch, Green, Griffiths, Müller-Stach, Murre, Voisin and others.

We show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension 8. We compare its Satake-Baily-Borel compactification with a compactification obtained by means of geometric invariant theory, considered by Miranda.