Adic Modular Forms

Adic 模块化形式

• 批准号: RGPIN-2016-06731
• 负责人: Iovita, Adrian
• 金额: $ 3.35万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654292
• 关键词: Adic; Modular; Forms

The research projects that I would like to propose are projects in the field of Number Theory. Number Theory is one of the oldest branches of Mathematics, its origins in the Western civilization can be traced back to antiquity. Its main object of study is the set of integer numbers which were seen by the disciples of the Pythagorean school as magical carrying in mysterious ways all the information in existence.***After two thousand years if intense study the set of integer numbers is still mysterious and very attractive for human enquiry. During the seventeenth century (AD 1637) Pierre de Fermat started investigating the integer solutions of the family of equations X^N+Y^N=Z^N, for N=3,4,5,.... This study is known as ``Fermat's last theorem" and it states that: there are no non-zero integer solutions to the equations X^N+Y^N=Z^N, for N=3,4,5,...***Fermat thought he proved the theorem but did not write the solution. Attempts to recover Fermat's solution (assuming it ever existed)***or to otherwise substantiate his claims spawned the birth of new Mathematical theories to which many great mathematicians contributed in profound ways culminating with Andrew Wiles' first published proof of the Theorem in 1995. Wiles' proof of Fermat's last theorem parlays a non-trivial solution of the ***equation X^p+Y^p=Z^p into the construction of an elliptic curve which can be then shown to posses an unlikely assortment of properties; he was then able to ***show that such an elliptic curve cannot exist by exploiting a deep, far reaching and still largely unproved connection between the various algebraic structures that arise from ``modular forms". Their role in our subject is fundamental and they, modular forms that is, are also the main characters of my own research. More precisely I am interested in understanding the $p$-adic properties of modular forms. Having fixed a prime integer p, i.e. any integer from the (infinite) list: 2,3,5,7,11,13,17,19,23,... it is interesting to study the divisibility of various integers by powers of p and this can be seen as defining a ``new distance" on the set of integers: namely two integers are ``near" if their***difference is divisible by a high power of p. In this funny distance, called p-adic distance, there are many gaps between various integers and if we fill in all these gaps, we obtain a much larger set (ring) called the ring of p-adic integers. The p-adic geometry is the study of p-adic solutions of polynomial equations with p-adic coefficients.***The $p$-adic world is entirely different from the world we see outside our window but this does not make it less real or interesting. In one of my research projects I propose to study the p-adic properties of modular forms and the geometry of the p-adic space, called eigencurve, which parameterizes the p-adic modular eigenforms (overconvergent)***of finite slope and its natural boundary.. *** **

我想提出的研究项目是数论领域的项目。数论是数学最古老的分支之一，它在西方文明中的起源可以追溯到古代。它的主要研究对象是一组整数，这些整数被毕达哥拉斯学派的信徒视为以神秘的方式携带着所有存在的信息。经过两千年的深入研究，这组整数仍然是神秘的，对人类的探索非常有吸引力。在17世纪（公元1637年），皮埃尔·德·费马开始研究方程组X^N+Y^N=Z^N的整数解，其中N=3,4,5,....这项研究被称为“费马大定理”，它指出：对于方程X^N+Y^N=Z^N，对于N=3,4,5，…，没有非零整数解。费马认为他证明了定理，但没有写出解。试图恢复费马的解（假设它曾经存在）***或以其他方式证实他的主张催生了新的数学理论的诞生，许多伟大的数学家以深刻的方式贡献了安德鲁·怀尔斯在1995年首次发表的定理证明。怀尔斯对费马大定理的证明将方程X^p+Y^p=Z^p的非平凡解运用到椭圆曲线的构造中，然后可以证明椭圆曲线具有一系列不太可能的性质；然后，他能够利用由“模形式”产生的各种代数结构之间的深刻的、深远的、很大程度上尚未被证明的联系，证明这样的椭圆曲线不可能存在。它们在我们的课题中的作用是基础性的，它们，模块形式，也是我自己研究的主要特征。更确切地说，我感兴趣的是理解模形式的$p$-adic属性。有一个固定的素数整数p，即（无限）列表中的任意整数：2,3,5,7,11,13,17,19,23，…有趣的是研究各种权力的整数p的可分性,这可以看作是定义一个“新的距离“整数的集合:即两个整数“附近”如果他们的* * *的区别是整除的高功率p。在这个有趣的距离,称为p进距离,有许多不同的整数之间的差距,如果我们填补这些空白,我们得到一个更大的组称为p进的环(环)整数。p进几何是研究具有p进系数的多项式方程的p进解。*** $p$-adic的世界与我们窗外看到的世界完全不同，但这并不会降低它的真实性或趣味性。在我的一个研究项目中，我提议研究模形式的p进性质和p进空间的几何，称为特征曲线，它参数化有限斜率及其自然边界的p进模特征形式（过收敛）***。* * * * *