Local Szpiro Conjecture and Congruences Between Modular Forms

20世纪80年代，Lucien Szpiro提出了一个猜想，该猜想与一条给定椭圆曲线上的两个不变量的大小有关。在这个公式之后不久，很明显，他的猜想与丢番图方程中另一个非常深的猜想密切相关，称为ABC猜想。这些猜想如果属实，将在数论中产生许多有趣的结果。例如，它们可以为费马大定理及其许多推广提供一个非常简短的证明。自从利用模性和水平降阶的结果证明费马大定理以来，数学界已经用类似的技术证明了费马大定理的许多推广。人们可以把所有这些都看作是证明Szpiro猜想和ABC猜想正确的证据。然而，目前还不清楚怀尔斯的模块化机械是否能像所说的那样说明斯兹皮罗猜想。我的研究包括研究一个猜想，称为局部Szpiro猜想，它更适合于模块化机械，同时仍然有许多有趣的应用于丢番图方程。特别地，这个猜想给出了某些模形式之间同余的精确界。这些界可以用来证明费马大定理的许多推广。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.