Descent for the punctured universal elliptic curve, and the average number of integral points on elliptic curves
arXiv:1502.04923
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 . 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 -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 .
22 pages, 2 tables. Major changes include: The statement of the main theorem was incorrect in the previous version. Section 7 is removed. Introduction is rewritten