# A Constructive Proof of Masser's Theorem

## TL;DR

This paper provides a constructive proof demonstrating the infinite existence of certain elliptic curves, called good Frey curves, related to the Modified Szpiro Conjecture and the abc Conjecture.

## Contribution

It offers a constructive proof that there are infinitely many good Frey elliptic curves, advancing understanding of the Modified Szpiro Conjecture.

## Key findings

- Constructive proof of infinite good Frey curves
- Supports the Modified Szpiro Conjecture
- Links to the abc Conjecture

## 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\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.