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

TL;DR

This paper establishes an upper bound on the average number of integral points on elliptic curves using a descent map and various number-theoretic bounds, providing new insights into the distribution of integral points.

Contribution

The paper introduces a novel descent map for elliptic curves that bounds the average number of integral points, extending classical methods with new bounds and estimations.

Findings

Average number of integral points bounded by 2.1 x 10^8

Descent map associates binary forms to integral points

Method extends to S-integral points with limitations

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 $S=\{2\}$.