Mathematical Sciences: L-value Congruences, Galois Structureand Arithmetic Surfaces

本研究将主要围绕两个方面展开。第一个问题是关于乘法伽罗瓦结构理论所预言的L函数的值的同余。第二个问题是关于阿拉克洛夫理论和阿德勒能力理论之间的联系。还将继续确定具有给定尖点数目的最小算术双曲三流形，并研究具有给定尖点数目的所有双曲三流形是否通过算术流形实现最小体积。最后，研究了整数环的马勒测度、L级数和K-群之间的关系。这项研究将涵盖数论和相关领域的广泛领域。它将进入分析、代数和几何领域。所有这些领域之间的联系将集中在赋予这项研究特别的吸引力上。这些努力将带来许多极大的兴趣。

This research will focus mainly on two areas. The first concerns congruences for values of L-functions predicted by the theory of multiplicative Galois structures. The second concerns the connections between Arakelov theory and adelic capacity theory. Work will also continue on determining the smallest arithmetic hyperbolic three manifold with a given number of cusps and to study whether the minimal volume of all hyperbolic three manifolds with a given number of cusps is achieved by an arithmetic manifold. Finally, the connections between Mahler measures, L-series and K-groups of rings of integers will be studied. This research will encompass broad areas of number theory and related areas. It will move into areas of analysis, algebra and geometry. The connections between all these areas will be focused upon giving this research a particular appeal. Much of great interest will come from these endeavors.