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的一个重要而活跃的研究领域。Stak的主要猜想提供了Artin L-函数的特殊值与相关Galois扩张的算法之间的联系。近年来，鲁宾和波佩斯库在阿贝尔L的情况下，对斯塔克的主要猜想--在原点消失的任意级函数--提出了精化的、完整的版本。此外，Burns和Flach通过对早期的Bloch-Kato和Fontaine-Perrin Riou猜想的修正，提出了某些动机L函数类的等变Tamagawa数猜想。如果局限于Artin L函数的情形，Burns-Flach猜想也可以被视为斯塔克主要猜想的精炼的、完整的版本。主要研究人员致力于为鲁宾猜想、波佩斯库猜想和伯恩斯-弗拉赫猜想提供证据、研究其作用行为，并寻找它们之间的联系。他还致力于开发这些语句的大体型p-ary精化，以及在这些语句与欧拉系统理论、等变岩泽理论以及Brumer、Leopoldt和Chinburg猜想之间建立桥梁。L函数是分析(连续)性质的数学对象，编码了大量算术(离散)性质的极其有趣和有用的信息。这个项目的主要目标是继续一个由斯塔克、鲁宾、首席调查员和伯恩斯-弗拉赫发起的计划，并开发旨在检索以特殊类型的L-函数(Artin L-函数)编码的算术数据的通用配方(猜想)，并在几个重要的特殊情况下遵循这些配方(换句话说，证明这些猜想)。同时，首席研究人员正在发展这些猜想与其他已经发展的理论之间的联系，处理算术(离散)性质的对象，如欧拉系统理论和等变岩泽理论。除了对称为数论和算术代数几何的纯数学核心领域的重要性外，这项研究还可能对开发新的数据加密算法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.