Iwasawa Theory and Arithmetic Geometry

阿格布拉9700937该奖项将资助算术代数几何中两个主题的研究。第一部分是岩泽理论和L函数。PI将研究Birch和Swinnerton-Dyer猜想的新方法以及p-进L函数理论。这些方法起源于算术伽罗华模结构理论，它们导致了研究交换簇和p-进表示的岩泽理论的新方法。第二个主题涉及有限域上的等变代数几何和高维簇的K-理论。PI将研究品种上等变向量丛的简化Grothendieck群。他将以正特征发展局部自由类群的几何，并将探索与高维类场理论和晶体上同调的联系。本方案所描述的研究属于算术代数几何领域。这是一门融合了两个最古老的数学分支-数论和几何--的学科，它已经蓬勃发展到解决了几个世纪以来一直存在的问题。它在物理、机器人、数据处理和信息理论等领域都有应用。

Agboola 9700937 This award will fund research on two topics in arithmetic algebraic geometry. The first is concerned with Iwasawa theory and L-functions. The PI will study new approaches to the Birch and Swinnerton-Dyer conjecture and the theory of p-adic L-functions. These approaches have their origins in the theory of arithmetic Galois module structure, and they lead to new ways of studying the Iwasawa theory of abelian varieties and p-adic representations. The second topic deals with equivariant algebraic geometry and the K-theory of higher dimensional varieties over finite fields. The PI will study reduced Grothendieck groups of equivariant vector bundles on varieties. He will develop the geometry of locally free class groups in positive characteristic, and he will explore connections with higher-dimensional class field theory and crystalline cohomology. The research described in this proposal lies in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest branches of mathematics---number theory and geometry---and which has blossomed to the point where it has solved problems that have stood for centuries. It finds applications in fields as diverse as physics, robotics, data processing and information theory.