1-Motives, Equivariant Iwasawa Theory and Special Values of L-functions

全局和p进l函数的特殊值理论在数论中具有核心意义。它在数论的解析方面和算术-代数-几何方面之间建立了微妙的联系，对两者都有了新的认识，并为数论中突出的开放问题提供了引人注目的解决方案。PI开发了一个推测程序，推广和改进了关于非正整数上全局l函数的特殊值的Gross- rubin - stark猜想和coats - sinnott猜想，推广和改进了Iwasawa理论中的主要猜想和Gross关于非正整数上p进l函数的特殊值的猜想。PI将运用i -动机的等变Iwasawa理论及其l-进实现的Iwasawa理论类似物、Weil-etale上同调、晶体上同调和t-动机理论为他的猜想提供新的证据。他还将尝试在他的猜想程序与Burns-Flach（关于arn动机的等变Tamagawa数猜想）和Ritter-Weiss（关于Iwasawa理论中主猜想的非阿贝等变版本）的猜想程序之间建立联系。PI还将研究他的推测程序在显式欧拉系统的构建、在任意全局域上显式生成类域（希尔伯特的第10个问题）和精炼类数公式方面的应用。PI将提供研究生课程，并组织由l函数特殊值领域的学生和专家参加的研讨会和会议。l函数是一种解析（连续）性质的小工具，它编码了大量的算术-代数-几何（离散）数据，这些数据对数论和算术-代数几何领域的专家以及密码学家、编码理论家、电信工程师等感兴趣。pi猜想程序建立在Gross， Rubin， Stark和Iwasawa等人的经典猜想的基础上，旨在确定（解码）实轴上积分点上l函数的特殊值的离散数据。PI将采用来自数学各个领域的技术，特别是数论和算术几何，在几个重要的案例中证明他的猜想，并在他的猜想程序与Burns-Flach和Ritter-Weiss的猜想程序之间建立联系。除了它在数论和代数几何方面的许多深远应用（PI将对此进行探索）之外，该研究项目对密码学和编码理论等实际领域也有重大的潜在影响。

The theory of special values of global and p-adic L-functions is of central importance in number theory. It establishes subtle links between the analytic aspects of the theory and the arithmetic-algebraic-geometric aspects, shedding new light on both and leading to striking solutions to outstanding open problems in number theory. The PI has developed a conjectural program which generalizes and refines the Gross-Rubin-Stark Conjectures and the Coates-Sinnott Conjectures on special values of global L-functions at non-positive integers and generalizes and refines the Main Conjecture in Iwasawa Theory and Gross's Conjectures on special values of p-adic L-functions at non-positive integers. The PI will employ techniques of equivariant Iwasawa Theory of 1-motives and the Iwasawa theoretic analogues of their l-adic realizations, Weil-etale cohomology, crystalline cohomology and the theory of t-motives to provide new evidence for his conjectures. He will also attempt to establish links between his conjectural program and those of Burns-Flach (on the Equivariant Tamagawa Number Conjecture for Artin Motives) and Ritter-Weiss (on non-abelian equivariant versions of the Main Conjecture in Iwasawa Theory). The PI will also study applications of his conjectural program to the construction of explicit Euler Systems, explicit generation of class-fields over an arbitrary global field (Hilbert's 10th problem) and refined class-number formulas. The PI will offer graduate courses and organize seminars and conferences bringing together students and experts in the field of special values of L-functions.An L-function is a gadget of analytic (continuous) nature which encodes a tremendous amount of arithmetic-algebraic-geometric (discrete) data of interest to experts working in the fields of number theory and arithmetic-algebraic geometry, as well as cryptographers, coding theorists, telecommunications engineers etc. The PIs conjectural program builds upon classical conjectures due to Gross, Rubin, Stark and Iwasawa among others and aims for determining (decoding) the discrete data out of special values of L-functions at the integral points on the real axis. The PI will employ techniques coming from various areas of mathematics, especially number theory and arithmetic geometry to prove his conjectures in several significant cases and to establish links between his conjectural program and those of Burns-Flach and Ritter-Weiss. Aside from its many far reaching applications to number theory and algebraic geometry (which will be explored by the PI), this research project has significant potential impact upon practical fields such as cryptography and coding theory.