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

1-动机、等变岩泽理论和 L 函数的特殊值

• 批准号: 0901447
• 负责人: Cristian Popescu
• 金额: $ 16.2万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901447&HistoricalAwards=false
• 关键词: Motives; Equivariant; Iwasawa; Theory; Special

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.

整体和p-adic L-函数的特殊值理论在数论中具有重要意义。它建立了微妙的联系之间的分析方面的理论和算术代数几何方面，脱落新的光都导致引人注目的解决方案，突出开放的问题，数论。PI已经开发了一个数学程序，该程序推广和改进了关于全局L-函数在非正整数处的特殊值的Gross-Rubin-Stark猜想和Coates-Sinnott猜想，并推广和改进了Iwasawa理论中的主要猜想和关于p进L-函数在非正整数处的特殊值的Gross猜想。PI将采用1-动机的等变岩泽理论和它们的l-adic实现的岩泽理论类似物，Weil-Etale上同调，结晶上同调和t-动机理论的技术，为他的理论提供新的证据。他还将试图建立他的数学程序之间的联系和伯恩斯-Flach（对等变玉川数猜想的阿丁动机）和里特-韦斯（非阿贝尔等变版本的主要猜想在岩泽理论）。PI还将研究他的数学程序在构建显式欧拉系统，在任意全局域上显式生成类域（希尔伯特第10问题）和细化类数公式中的应用。PI将提供研究生课程，并组织研讨会和会议，将学生和专家聚集在L-函数的特殊值领域。（连续）性质，编码了大量的算术-代数-几何（离散）数据感兴趣的专家在数论和算术代数几何领域的工作，以及密码学家，编码理论家，该PI算法程序建立在Gross、Rubin、Stark和Iwasawa等人的经典算法之上，旨在确定（解码）真实的轴上积分点处L函数特殊值的离散数据。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，以证明他在几个重要情况下的理论，并建立他的数学程序与Burns-Flach和Ritter-Weiss之间的联系。除了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，该研究项目还对密码学和编码理论等实际领域具有重大的潜在影响。