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

这项研究将集中在关于上同调群上的某些体积形式的Faltings定义的思想，这些思想被用于他对Arakelov曲面的Riemann-Roch定理的证明。首席调查者发现了一种新的方法来研究这些体积形式，从上同调群上的某些度量中获得它们。他对这些度量有一个相当直接的几何结构，这表明它们类似于通过选择模型而在有限位置诱导的非阿基米德度量。他将把这种构造应用于Riemann曲面的Faltings增量不变量的研究，该不变量的非阿基米德模拟的定义和研究，以及Arakelov的Zeta函数的研究。这项研究是在应用几何方法来求解整数方程(丢番图方程)的领域。最近在这一领域的基础工作是这位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.