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

Volume 183 / 2018

Dohyeong Kim Acta Arithmetica 183 (2018), 201-222 MSC: 11D, 11G05, 11D59. DOI: 10.4064/aa8458-11-2017 Published online: 27 April 2018


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 back at least 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 a numerical example with $S=\{2\}$.


  • Dohyeong KimDepartment of Mathematics
    University of Michigan
    2074 East Hall
    530 Church Street
    Ann Arbor, MI 48109-1043, U.S.A.

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