A constructive proof of Masser’s Theorem

Alexander J. Barrios
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引用次数: 2

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\} $. Masser showed that there are infinitely many good Frey curves. Here we give a constructive proof of this assertion.
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马瑟定理的构造性证明
与$abc$猜想等价的修正斯皮罗猜想指出,对于每个$\epsilon>0$,存在有限多条满足$N_{E}^{6+\epsilon}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $的有理椭圆曲线,其中$c_{4}$和$c_{6}$是与$E$的最小模型相关的不变量,$N_{E}$是$E$的导体。我们说$E$是一条好的椭圆曲线,如果$N_{E}^{6}<\max\!\left\{ \left\vert c_{4}^{3}\right\vert,c_{6}^{2}\right\} $。Masser证明了有无限多条好的Frey曲线。这里我们对这个论断给出建设性的证明。
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