Arithmetic Invariants and Their Non-Triviality

这个项目涉及数论的研究，这是一个与密码学和物理学等更多应用领域有许多有趣联系的学科。数学家通过附加不变量来研究算术对象，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它相关的某些数学对象，称为它的L函数。由于它们是复变量或p元变量的函数，人们可以用整数来计算它们，从而得到与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.找到两个或多个不同性质的算术不变量之间的关系(或恒等式)；b.区分重要算术不变量的非平凡和平凡。这个项目将发展一个处理问题B的代数理论。特别是，本研究将研究Zeta函数及其导数的值的非平凡。通过一个众所周知的原理(例如，Birch-Swinnerton Dyer猜想)，这些L值编码了深丢番图问题的解，如果我们能证明Zeta值不为零或为零，我们应该能够预测模方程和椭圆型方程有多少有理解。本研究旨在发展一套系统的理论来区分重要算术不变量的非平凡与平凡。这位研究人员和一位合作者的早期工作发展了对p-进Galois表示和Hecke代数、p-进解析族的模形式及其L函数以及算术不变量分析的新理解。本研究项目将对这些进展进行跟踪。例如，该项目将研究如何利用大Hecke代数中的系数来测量模Galois表示的图像的大小(关于图像的一个非平凡问题)。作为另一个例子，这项工作将研究p元L函数的非零性(和模a素数的非零性)，以及在有理椭圆曲线和阿贝尔簇中创建有理点的模尝试的非平凡性。

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.