A Constructive Proof of Masser's Theorem

Abstract

The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $\epsilon >0$, there are finitely many rational elliptic curves satisfying $N_E^{6+\epsilon }<\max \left\{\left|c_4^3\right|,c_6^2\right\}$ where $c_4$ and $c_6$ are the invariants associated to a minimal model of $E$ and $N_E$ is the conductor of $E$. We say $E$ is a good elliptic curve if $N_E^6<\max \left\{\left|c_4^3\right|,c_6^2\right\}$. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.

arXiv comment9 pages