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

这项研究将集中在周围的想法 上同调群上某些体积形式的faltings定义， 这是他在证明黎曼-罗克定理时用到的， 阿拉克洛夫浮出水面 首席调查员发现了一个 新的方法，这些体积的形式，获得他们从某些 上同调群上的度量 他有一个相当直接的 这些指标的几何结构表明它们在 与非阿基米德度量的类比， 选择一个模型。 他将应用这个结构 对于黎曼曲面的Faltings δ不变量的研究， 对非阿基米德类比的定义和研究 不变量，并研究阿拉克洛夫的zeta函数。 本研究属于几何方法应用领域 求解整数方程（丢番图方程）。 法尔廷斯 最近在这一领域的基础工作是出发点， 这位研究员的工作。

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.