Stark-type Conjectures "over Z" and the Equivariant Tamagawa Number Conjecture

l函数的特殊值理论是数论和算术代数几何中一个重要的、活跃的研究领域。Stark的主猜想提供了一个在Artin l函数的特殊值和相关伽罗瓦扩展的算法之间的联系。近年来，Rubin和Popescu在任意阶的阿贝尔l函数在原点消失的情况下，提出了Stark主要猜想的改进的、积分的版本。此外，Burns和Flach通过对Bloch-Kato和fontaine - perrin Riou的早期猜想进行再加工，为某些类别的动机l函数制定了等变Tamagawa数猜想。如果限制在Artin l -函数的情况下，Burns-Flach猜想也可以看作是Stark主猜想的一个改进的、完整的版本。首席研究员的重点是为鲁宾、波佩斯库和伯恩斯-弗拉赫的猜想提供证据，研究它们的功能行为，并找到它们之间的联系。他还致力于发展这些陈述的gross -type p-adic改进，并在这些陈述与欧拉系统理论、等变Iwasawa理论以及Brumer、Leopoldt和chinburg的猜想之间建立桥梁。l函数是解析（连续）性质的数学对象，编码了大量非常有趣和有用的算术（离散）性质的信息。该项目的主要目标是继续Stark， Rubin，首席研究员和Burns-Flach发起的计划，并开发旨在检索特殊类型l -函数（Artin l -函数）中编码的算术数据的一般食谱（猜想），并在几个重要的特殊情况下遵循这些食谱（换句话说，证明这些猜想）。与此同时，首席研究员正在发展这些猜想与其他已经发展的理论之间的联系，这些理论处理算术（离散）性质的对象，如欧拉系统理论和dequivariant Iwasawa理论。除了对数论和算术代数几何等纯数学核心领域的重要性外，这项研究在开发新的数据加密算法方面可能具有深远的实际应用。

The theory of special values of L-functions is a major, active area ofresearch within the general fields of number theory and arithmeticalgebraic geometry. Stark's Main Conjecture provides a link betweenspecial values of Artin L-functions and the arithmetic of the associatedGalois extensions. In recent years, Rubin and Popescu have formulatedrefined, integral versions of Stark's Main Conjecture in the case ofabelian L-functions of arbitrary order of vanishing at the origin. Also,Burns and Flach, by reworking earlier conjectures of Bloch-Kato andFontaine-Perrin Riou, have formulated the Equivariant Tamagawa NumberConjecture for certain classes of motivic L-functions. If restricted tothe case of Artin L-functions, the Burns-Flach conjecture can also beviewed as a refined, integral version of Stark's Main Conjecture. ThePrincipal Investigator focuses on providing evidence for, studying thefunctorial behavior of, and finding links between the Conjectures ofRubin, Popescu, and Burns-Flach. He also works on developingGross-type p-adic refinements of these statements, as well as buildingbridges between these statements and the Theory of Euler Systems,Equivariant Iwasawa Theory, and the Conjectures of Brumer, Leopoldt, andChinburg.The L-functions are mathematical objects of analytic (continuous) nature,encoding an enormous amount of extremely interesting and usefulinformation of arithmetic (discrete) nature. The main goal of thisproject is to continue a program initiated by Stark, Rubin, the principalinvestigator, and Burns-Flach, and develop general recipes (conjectures)aimed at retrieving the arithmetic data encoded in a special type ofL-functions (the Artin L-functions), and follow these recipes (in otherwords prove these conjectures) in several important special cases. Inparallel, the Principal Investigator is developing links between theseconjectures and other, already developed theories, dealing with objects ofarithmetic (discrete) nature, such as the theory of Euler Systems andEquivariant Iwasawa Theory. Aside from its importance for the centralareas of pure mathematics called number theory and arithmetic algebraicgeometry, this research could have far reaching practical applications tothe development of new data encryption algorithms.