Special Values of p-adic L-Functions

p 进 L 函数的特殊值

• 批准号: 1600943
• 负责人: Samit Dasgupta
• 金额: $ 15.9万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600943&HistoricalAwards=false
• 关键词: Special; Values; adic; Functions

One of the central themes in modern algebraic number theory is that local information about a mathematical structure can often be pieced together in surprising ways to yield global information about the structure. The passage from local to global in number theory is deep, mysterious, and not well-understood. This general philosophy can be made concrete through precise conjectures on values of L-series, which are certain functions defined in terms of the local behavior of mathematical structures. These conjectures assert that the values of L-series at integers reveal global arithmetic invariants of the structures. While there are plentiful and very general conjectures regarding special values of L-series, only special cases of the conjectures have been proven. This research project aims to generalize these results and deepen understanding in this important area of number theory.Some of the most important and well-known conjectures in number theory concern the special values of L-series, including the conjectures of Birch and Swinnerton-Dyer, Stark, Beilinson, and Bloch and Kato. For example, the Birch and Swinnerton-Dyer Conjecture predicts that the value at s=1 of the L-series defined in terms of the number of points on an elliptic curve over the integers modulo p for all primes p carries information on the size of the set of points of the elliptic curve over the rational numbers. In many cases, the L-series in question have alternate manifestations known as p-adic L-series, which are defined by applying a different notion of analysis from that encountered in standard Euclidean geometry. Special-value formulae for classical and p-adic L-series hold an important place at the center of modern number theory research. In this project, the investigator will continue to study conjectures on the special values of p-adic L-functions. The methods involve studying certain explicit families of Eisenstein series and build on work on the Iwasawa Main Conjecture. The project aims to develop new techniques to prove p-adic special-value conjectures, as well as various generalizations and refinements.

现代代数数论的中心主题之一是，关于数学结构的局部信息通常可以以令人惊讶的方式拼凑在一起，以产生关于结构的全局信息。 数论中从局部到全局的过程是深刻的、神秘的，而且还没有被很好地理解。 这一普遍的哲学可以通过精确地描述L-级数的值来具体化，L-级数是根据数学结构的局部行为定义的某些函数。 这些定理断言L-级数在整数处的值揭示了其结构的全局算术不变量。 虽然关于L-级数的特殊值有很多非常一般的证明，但只有特殊情况的证明才被证明。 本研究计划旨在推广这些结果，加深对数论这一重要领域的理解。数论中一些最重要和最著名的理论涉及L-级数的特殊值，包括Birch和Swinnerton-Dyer，Stark，Beilinson，Bloch和Kato的理论。 例如，Birch和Swinnerton-Dyer猜想预测，对于所有素数p，根据整数模p上椭圆曲线上的点的数量定义的L级数在s=1处的值携带关于有理数上椭圆曲线的点集的大小的信息。 在许多情况下，所讨论的L-级数有称为p进L-级数的替代表现，其定义是通过应用与标准欧几里得几何中遇到的不同的分析概念。 古典和p-adic L-级数的特值公式在现代数论研究中占有重要的地位。在本项目中，研究者将继续研究p-adic L-函数的特殊值。该方法涉及研究某些显式家庭的爱森斯坦系列和建设的工作岩泽主要猜想。 该项目旨在开发新的技术来证明p-adic特殊值构造，以及各种推广和改进。