This research will concentrate on the ideas surrounding Faltings definition of certain volume forms on cohomology groups, which were used in his proof of the Riemann-Roch theorem for Arakelov surfaces. The Principal Investigator has discovered a new approach to these volume forms, obtaining them from certain metrics on the cohomology groups. He has a rather direct geometric construction of these metrics which shows them to be in analogy with the non-archimedean metrics induced at the finite places by the choice of a model. He will apply this construction to the study of Faltings' delta invariant for Riemann surfaces, to the definition and study of the non-archimedean analog of this invariant , and to the study of Arakelov's zeta functions. This research is in the area of applying geometric methods to solve equations in integers (Diophantine equations). Faltings recent fundamental work in this area is the jumping off point for the work of this researcher.
