Local Szpiro Conjecture and Congruences Between Modular Forms

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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.