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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710376
• 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)&n

我想提出的研究项目是数论领域的项目。数论是数学中最古老的分支之一，它的起源可以追溯到古代的西方文明。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弟子视为以神秘的方式携带所有存在的信息。 经过两千年的深入研究，整数集合仍然是神秘的，对人类的研究非常有吸引力。 在17世纪（公元1637年），皮埃尔·德·费马开始研究方程族X^N+Y^N=Z^N的整数解，其中N= 3，4，5，.。这项研究被称为“费马最后定理”，它指出：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，但没有写出解答。试图恢复费马的解决方案（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，许多伟大的数学家作出了贡献，以深刻的方式最终与安德鲁怀尔斯'第一次发表证明定理在1995年。怀尔斯对费马最后定理的证明， 方程X^p+Y^p=Z^p转化为椭圆曲线的构造，然后可以证明椭圆曲线具有一种不太可能的性质;然后他能够 表明，这样一个椭圆曲线不能存在，通过利用一个深，深远的，仍然在很大程度上未经证明的各种代数结构之间的联系，从“模块形式”。它们在我们的学科中的作用是基本的，它们，即模块形式，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性质。[2019 - 03 - 19 00：01：00][2019 - 03 - 19 00：00：00][2019 - 03 - 19 00：00][2019 - 01：00][2019 - 01：00]有趣的是，研究各种整数被p的幂整除，这可以被看作是在整数集合上定义一个"新距离”：即两个整数是"接近”的，如果它们的 在这个有趣的距离中，称为p-adic距离，各种整数之间存在许多间隙，如果我们填充所有这些间隙，我们会得到一个更大的集合（环）&n