Adic Modular Forms

我要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，它起源于西方文明，可以追溯到古代。它的主要研究对象是整数集，它被毕达哥拉斯学派的信徒视为神奇地以神秘的方式承载着所有存在的信息。*经过两千年的密集研究，整数集仍然神秘，非常吸引人类的探究。在17世纪(公元1637年)，皮埃尔·德·费马开始研究方程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不存在非零整数解，对于N=3，4，5，……*费马认为他证明了这个定理，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，许多伟大的数学家以深刻的方式对此做出了贡献，安德鲁·怀尔斯于1995年首次发表了《定理的证明》。Wiles对Fermat最后定理的证明将*方程X^p+Y^p=Z^p的非平凡解应用到椭圆曲线的构造中，然后可以证明该椭圆曲线具有不太可能的各种性质；然后，他能够*证明这样一条椭圆曲线不可能存在，这是通过利用从‘模形式’产生的各种代数结构之间的深刻、深远且在很大程度上仍未得到证明的联系来实现的。它们在我们的学科中的作用是基本的，它们，也就是模块形式，也是我自己研究的主要特征。更确切地说，我感兴趣的是了解模形式的$p$-进的性质。有固定的素数p，即(无限)列表中的任何整数：2，3，5，7，11，13，17，19，23，...用p的幂研究各种整数的可除性是很有趣的，这可以被看作是在整数集上定义了一个新的距离：即，如果两个整数的*差可以被p的高次幂整除，则两个整数是“接近”的。在这个有趣的距离中，称为p-adi距离，在不同的整数之间有许多间隙，如果我们填满所有这些间隙，我们得到一个更大的集合(环)，称为p-adi整数环。P元几何是研究具有p元系数的多项式方程的p元解。*p元元世界与我们在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，我建议研究模形式的p-Add性质和p-Add空间的几何，称为本征曲线，它将有限斜率及其自然边界的p-ady模特征形式(超收敛)*参数化。*

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.. *** **