Abstract
The Frey curve y2=x( x+a) ( x-b) is a two-parameter family of elliptic curves which comes equipped with an easily computable minimal discriminant which helped pave the mathematical bridge that led to the proof of Fermat's Last Theorem. In this dissertation, we extend the ideas of the Frey curve by considering two- and three- parameter families of elliptic curves which parameterize all rational elliptic curves with non-trivial torsion. First, we use these families to give a new proof of a classic result of Frey, Flexor, and Oesterle which pertains to the primes at which an elliptic curve over a number field can have additive reduction. While our proof gives a weaker variant of the original statement, it is explicit and does not require the Neron model of an elliptic curve. As a consequence of this new proof, we attain our classification of minimal discriminants of rational elliptic curves with non-trivial torsion. In addition, we give necessary and sufficient conditions for when a rational elliptic curve with non-trivial torsion has additive reduction at a given prime. We also study the connection between torsion structure of a rational elliptic curve and the possible reduced minimal models.
The second theme of this dissertation concerns the modified Szpiro conjecture, which is equivalent to the ABC Conjecture. Roughly speaking, the modified Szpiro conjecture states that certain elliptic curves, known as good elliptic curves, are rare in nature. Masser gave a non-constructive proof which showed that there were infinitely many good Frey curves. In this dissertation, we give a constructive proof of Masser's assertion. We then extend this result by proving that for each of the fifteen torsion subgroups T allowed by Mazur's Torsion Theorem, there are infinitely many good elliptic curves E with torsion subgroup isomorphic to T. This proof is also constructive and allows for the construction of a database which consists of 13870964 good elliptic curves. We provide an analysis of these good elliptic curves to parallel the work done by the ABC Home project concerning the ABC Conjecture and good ABC triples. The data obtained is then used to formulate an explicit version of the modified Szpiro conjecture. We then show that this explicit formulation allows for the construction of databases of elliptic curves which are exhaustive up to a given conductor.
Lastly, we use the classification of minimal discriminants to study the local data of rational elliptic curves at a given prime via Tate's Algorithm. These results and a study of the naive height of an elliptic curve allow us to prove that there is a lower bound on the modified Szpiro ratio which depends only on the torsion structure of an elliptic curve.