• Corpus ID: 119312833

# The second moment of the number of integral points on elliptic curves is bounded

@article{Alpoge2018TheSM,
title={The second moment of the number of integral points on elliptic curves is bounded},
author={Levent Alpoge and Wei Ho},
journal={arXiv: Number Theory},
year={2018}
}
• Published 10 July 2018
• Mathematics
• arXiv: Number Theory
In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219 \ldots$, the $s$-th moment of the number of integral points is bounded for many families of elliptic curves --- e.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for…
3 Citations
Fix a non-square integer $k\neq 0$. We show that the number of curves $E_B:y^2=x^3+kB^2$ containing an integral point, where $B$ ranges over positive integers less than $N$, is bounded by $O_k(N(\log Between his arrival in Frankfurt in 1922 and and his proof of his famous finiteness theorem for integral points in 1929, Siegel had no publications. He did, however, write a letter to Mordell in 1926 We show that the total number of non-torsion integral points on the elliptic curves$\mathcal{E}_D:y^2=x^3-D^2x$, where$D$ranges over positive squarefree integers less than$N$, is$O( N(\log

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