Mathematical Sciences: A New Approach to Falting's Metrics on Cohomology in Arakelov Theory

本研究将集中于法尔廷对上同群上某些体积形式的定义，这些定义被用于他对Arakelov曲面的Riemann-Roch定理的证明。主要研究者发现了一种新的方法来获得这些体积形式，从上同调群上的某些度量中获得它们。他对这些度规有一个相当直接的几何构造，表明它们与非阿基米德度规相似，这些非阿基米德度规是通过选择一个模型在有限的地方产生的。他将把这种构造应用于研究Faltings黎曼曲面的delta不变量，定义和研究该不变量的非阿基米德类比，以及研究Arakelov的zeta函数。本研究的领域是应用几何方法求解整数方程（丢番图方程）。法尔廷最近在这一领域的基础性工作是本研究人员工作的起点。

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.