Mathematical Sciences: Higher Regulators, Algebraic Cycles, and Values of L-functions

本研究将继续P.I.关于圈的结构和算术簇的K理论及其与L函数在整点的取值的(猜想)关系的研究。他将在模簇的背景下分析贝林森、布洛赫和泰特的某些猜测，在模簇的背景下，人们可以使用自同构形式理论。这项研究涉及数论、代数几何和自同构形式等多个主题。也就是说，它利用代数、分析和几何的工具来分析数论问题。“L函数值”给出了具有重要算术意义的显式数的解析表达式，为深入研究提供了依据。P.I.在过去为这一领域贡献了许多有趣的成果，在本研究期间肯定会进一步这样做。

This research will continue the investigations of the P.I. on the structure of cycles and K-theory of arithmetic varieties and their (conjectural) relationship to values of L-functions at integral points. He will analyze certain selected conjectures of Beilinson, Bloch and Tate in the context of modular varieties where one can use the theory of automorphic forms. This research is in the blend of topics encompassing number theory, algebraic geometry and automorphic forms. That is, it brings to bear tools from algebra, analysis and geometry to analyze number theoretic problems. The "values of L-functions" give analytic expression to explicit numbers of great arithmetic significance allowing for their deeper study. The P.I. has contributed many interests results to this area in the past and will surely do so further during this research.