Senior Honors Thesis Presentation: Connections between the ABC Conjecture and the Modified Szpiro Conjecture

Abstract: The ABC Conjecture is one of the most important unsolved problems in Diophantine equations. A proof of the ABC Conjecture would lead to the solution of countless Diophantine equations such as Fermat’s Last Theorem. In 1975, Yves Hellegouarch began to associate solutions of Fermat’s equation (a, b, c) to a specific elliptic curve, the Frey Curve. The Frey Curve served as a counterexample to Fermat’s Last Theorem and provides a natural bridge between number theory and geometry. In 1988, a french mathematician, Lucien Szpiro, showed using the Frey Curve that the ABC Conjecture was equivalent to a conjecture by elliptic curves called the Modified Szpiro Conjecture. In the process, he proved the conjecture for certain special kinds of elliptic curves. In this thesis, we explore the connection between solutions of the ABC Conjecture, ABC triples, and solutions of the Modified Szpiro Conjecture, good elliptic curves. First, we construct new sequences on ABC triples and then show that the Modified Szpiro Conjecture holds for a certain isogeny classes of elliptic curves over Q.