Adic Modular Forms

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整数集合，毕达哥拉斯学派的弟子认为这些整数集合以神秘的方式承载着所有存在的信息。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，.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（假设它曾经存在过）* 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做出了贡献，最终在1995年安德鲁·怀尔斯首次发表了定理的证明。怀尔斯对费马最后定理的证明将 * 方程X^p+Y^p=Z^p的非平凡解转化为椭圆曲线的构造，然后可以证明椭圆曲线具有不太可能的各种性质;然后，他能够 * 表明，这样的椭圆曲线不可能存在，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的幂整除的可能性，这可以被看作是在整数集合上定义了一个"新距离”：也就是说，如果两个整数的 * 差可以被p的高次幂整除，那么它们就“接近”。在这个有趣的距离中，称为p-adic距离，各种整数之间有许多间隙，如果我们填充所有这些间隙，我们得到一个更大的集合（环），称为p-adic整数环。p-adic几何是研究具有p-adic系数的多项式方程的p-adic解。$p$-adic世界与我们在窗外看到的世界完全不同，但这并不会使它变得不那么真实的或有趣。在我的一个研究项目中，我建议研究模形式的p-adic性质和p-adic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-adic模特征形（过收敛）*。 *** **

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.. *** **