A Constructive Proof of Masser's Theorem

The Modified Szpiro Conjecture, equivalent to the $abc$ Conjecture, states that for each $ε>0$, there are finitely many rational elliptic curves satisfying $N_{E}^{6+ε}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\}$ 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 $N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\}$. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.

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