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

这项研究将主要集中在两个领域。 第一 涉及由L-函数预测的L-函数的值的同余， 乘法伽罗瓦结构理论 第二种情况是 阿拉克洛夫理论与阿拉克能力的关系 理论 还将继续努力确定最小的 具有给定尖点数的算术双曲三流形 并研究是否所有双曲三个最小体积 具有给定数量的尖点的流形是通过一个 算术流形 最后，马勒和 测度，整数环的L-级数和K-群将是 研究了 这项研究将涵盖数论的广泛领域 和相关领域。 它将进入分析，代数， 和几何学 所有这些区域之间的联系将是 专注于给这项研究一个特别的吸引力。 大部分 这些努力将引起极大的兴趣。

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.