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

### Volume 183 / 2018

#### 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 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 $S=\{2\}$.