Local Szpiro Conjecture and Congruences Between Modular Forms

在20世纪80年代，Lucien Szpiro提出了一个关于给定椭圆曲线上两个不变量大小的猜想。不久后，这一提法，很明显，他的猜想是非常密切相关的另一个非常深刻的猜想丢番图方程称为ABC猜想。这些假设如果是真的，将在数论中产生许多有趣的结果。例如，他们可以提供一个非常简短的证明费马的最后定理沿着与它的许多推广。自从费马最后定理的证明使用模块化和水平降低的结果，数学界已经应用类似的技术来证明费马最后定理的许多推广。人们可以把所有这些看作是斯皮罗猜想和ABC猜想有效性的证据。然而，目前还不清楚，如果怀尔斯的模块化机器可以说任何关于Szpiro猜想所述。我的研究涉及研究一个猜想，称为局部Szpiro猜想，它更适合于模块化机器，同时仍然有许多有趣的应用丢番图方程。特别是，这个猜想给出了一个精确的界限之间的某些模形式的同余。这些界限可以用来证明费马最后定理的许多推广。我建议使用模形式和算术几何的工具来研究局部Szpiro猜想。

In 1980s Lucien Szpiro formulated a conjecture relating the size of two invariants attached to a given elliptic curve. Shortly after this formulation, it became clear that his conjecture is very closely related to another very deep conjecture in Diophantine equations called the ABC conjecture. These conjectures, if true, will have many interesting consequences in number theory. For instance, they can provide a very short proof to Fermat's last theorem along with many of its generalizations. Since the proof of Fermat's last theorem using modularity and level lowering results, the mathematics community has applied similar techniques to prove many generalizations of Fermat's last theorem. One can view all these as evidence towards the validity of the Szpiro and the ABC conjecture. However it is unclear if Wiles's modular machinery can say anything about Szpiro conjecture as stated. My research involves studying a conjecture, called the local Szpiro conjecture, that is better suited for modular machinery, while still having many interesting applications for Diophantine equations. In particular, this conjecture gives a precise bound on congruences between certain modular forms. These bounds can be used to prove many generalizations of Fermat's last theorem. I propose to study the local Szpiro conjecture using tools from modular forms and arithmetic geometry.