Refined Conjectures

本项目支持Tate教授对l -函数特殊值的各种精细化猜想的研究，如Gross的猜想精细化类数公式和Stark猜想的相应精细化，2实数二次域的循环扩展中Stark单元的精细结构问题，3 5维Sklyanin代数。这项研究涉及数字和算术几何的一般领域。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。算术几何结合了数学中两个最古老的分支：数论和几何。这种结合产生的新见解正在产生越来越强大的工具来解决长期存在的问题，如费马大定理，它抵制了三个多世纪以来数学家的最强努力。此外，虽然算术几何有时被认为是纯数学中最纯粹的，但它也一直在发展富有洞察力的新技术，导致诸如纠错码和密码学等应用领域的巨大进步。

This project supports the research of Professor Tate on 1) various refined conjectures on special values of L-functions, such as Gross's conjectural refined class number formula and the corresponding refinements of Stark's conjectures, 2) questions about the fine structure of the Stark unit in a cyclic extension of a real quadratic field, and 3) Sklyanin algebras of dimension 5. This research lies in the general areas of number and arithmetic geometry. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems. Arithmetic geometry combines two of the oldest branches of mathematics: number theory and geometry. The new insights arising out of this combination are producing increasingly powerful tools to solve long-standing problems like Fermat's Last Theorem, which have resisted the strongest efforts of over three centuries of mathematicians. In addition, though Arithmetic Geometry is sometimes regarded as among the purest of pure mathematics, it has also been developing insightful new techniques leading to dramatic progress in such applied areas as error-correcting codes and cryptography.