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领域中一个重要而活跃的研究领域。 Stark的主要猜想提供了Artin L-函数的特殊值和相关Galois扩张的算术之间的联系。近年来，Rubin和Popescu在任意阶交换L-函数在原点为零的情况下，给出了Stark主猜想的精化的积分形式。此外，Burns和Flach通过修改Bloch-Kato和Fontaine-Perrin Riou的早期猜想，对某些类型的motivic L-函数建立了等变Tamagawa数猜想。如果仅限于Artin L-函数的情形，Burns-Flach猜想也可以看作是Stark主猜想的一个改进的积分版本。 主要研究者的重点是提供证据，研究的功能行为，并找到鲁宾，波佩斯库和伯恩斯-弗拉克猜想之间的联系。他还致力于发展格罗斯型p-adic细化这些报表，以及buildingbridges这些报表和理论的欧拉系统，Equivariant岩泽理论，和猜想的Brumer，Leopoldt，和Chinburg.The L-功能是数学对象的分析（连续）性质，编码了大量的极其有趣和有用的信息的算术（离散）性质。 这个项目的主要目标是继续由斯塔克，鲁宾，首席研究员，和伯恩斯-弗拉克发起的一个计划，并开发通用配方（算法），旨在检索编码在一种特殊类型的L-函数（阿廷L-函数）的算术数据，并遵循这些配方（换句话说，证明这些算法）在几个重要的特殊情况。与此同时，首席研究员正在开发thesconjectures和其他已经开发的理论之间的联系，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。除了对数论和算术代数几何这两个纯数学的中心领域的重要性之外，这项研究对开发新的数据加密算法也有着深远的实际应用。

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.