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arXiv · 1502.04923

Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves

Dohyeong Kim
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Abstract

We show that the average number of integral points on elliptic curves, counted modulo the natural involution on a punctured elliptic curve, is bounded from above by $2.1 \times 10^8$. To prove it, we design a descent map, whose prototype goes at least back to Mordell, which associates a pair of binary forms to an integral point on an elliptic curve. Other ingredients of the proof include the upper bounds for the number of solutions of a Thue equation by Evertse and Akhtari-Okazaki, and the estimation of the number of binary quartic forms by Bhargava-Shankar. Our method applies to $S$-integral points to some extent, although our present knowledge is insufficient to deduce an upper bound for the average number of them. We work out the numerical example with $S=\{2\}$.

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Dohyeong Kim. 2015-12-07. Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves. https://arxiv.org/abs/1502.04923

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