Arithmetic Invariants and Their Non-Triviality

算术不变量及其非平凡性

• 批准号: 1464106
• 负责人: Haruzo Hida
• 金额: $ 60万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-15 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1464106&HistoricalAwards=false
• 关键词: Arithmetic; Invariants; Their; Non; Triviality

This project concerns research in number theory, a subject that has many interesting connections to more applied areas such as cryptography and physics. Number theorists study arithmetic objects by attaching to them invariants in order to make them clearly visible. Each important object of interest in number theory has associated to it certain mathematical objects called its L-functions. As they are functions of complex or p-adic variables, one can evaluate them at integers, getting concrete numbers associated with the object under study. There are two fundamental problems concerning invariants in number theory:A. Find a relation (or an identity) among two or more arithmetic invariants of different nature;B. Distinguish between the non-triviality and triviality of important arithmetic invariants.This project will develop an algebraic theory dealing with Problem B. In particular, the research will study non-triviality of values of zeta functions and their derivatives. By a well-known principle (for example, the Birch-Swinnerton Dyer conjecture), these L-values encode solutions of deep Diophantine problems, and if we can show non-vanishing or vanishing of zeta values, we should be able to predict how many rational solutions modular and elliptic equations have. This research project aims to develop a systematic theory for distinguishing between the non-triviality and triviality of important arithmetic invariants. Earlier work of the investigator and a collaborator developed new understanding of p-adic Galois representations and Hecke algebras, p-adic analytic families of modular forms and their L-functions, and analysis of arithmetic invariants. This research project will follow on these developments. For example, the project will investigate how to measure the size of the image of modular Galois representation with coefficients in a big Hecke algebra (a nontriviality question concerning the image). As another example, the work will study non-vanishing (and non-vanishing modulo a prime) of p-adic L-functions and non-triviality of modular attempts of creating rational points in rational elliptic curves and abelian varieties.

该项目涉及数论研究，这一学科与密码学和物理学等更多应用领域有许多有趣的联系。数论学家通过给算术对象附加不变量来研究它们，以便使它们清晰可见。数论中每个重要的感兴趣的对象都与某些称为l函数的数学对象相关联。由于它们是复变量或p进变量的函数，人们可以将它们计算为整数，得到与所研究对象相关的具体数字。数论中关于不变量有两个基本问题：A。找出两个或多个不同性质的算术不变量之间的关系（或恒等）；区分重要的算术不变量的非平凡性和平凡性。这个项目将发展一个代数理论来处理问题b。特别是，研究将研究ζ函数及其导数值的非平凡性。根据一个众所周知的原理（例如，Birch-Swinnerton Dyer猜想），这些l值编码深丢芬图问题的解，如果我们能证明zeta值的非消失或消失，我们应该能够预测模块化和椭圆方程有多少理性解。本研究计划旨在建立一个区分重要算术不变量的非平凡性与平凡性的系统理论。研究者和合作者的早期工作发展了对p进伽罗瓦表示和Hecke代数，模形式的p进解析族及其l函数，以及算术不变量分析的新理解。这个研究项目将继续这些发展。例如，该项目将研究如何在大Hecke代数中测量带有系数的模伽罗瓦表示图像的大小（关于图像的非平凡性问题）。作为另一个例子，该工作将研究p进l函数的非消失（和非消失模a素数）和在有理椭圆曲线和阿贝尔变体中创建有理点的模尝试的非平凡性。