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

TL;DR

This paper proves that the second moment of the number of $S$-integral points on elliptic curves over a number field is bounded, introducing new bounds depending on the rank, class group, and primes dividing the discriminant.

Contribution

It provides new upper bounds on the number of $S$-integral points on elliptic curves, leading to bounded moments for families ordered by height and with marked points.

Findings

Second moment of $S$-integral points is bounded for elliptic curves over number fields.

Bound on integral points depends on rank, class group, and discriminant primes.

Bounded $r$-th moments for specific $r$ values in families of elliptic curves.

Abstract

Let $K$ be a number field and $S$ a finite set of places of $K$ containing all archimedean places. In this paper, we show that the second moment of the number of $S$-integral points on elliptic curves over $K$ is bounded. In particular, we prove that, for any positive real number $r\leq \log_2 5 = 2.3219 \ldots$, the $r$-th moment of the number of $S$-integral points is bounded for the family of all integral short Weierstrass curves ordered by height, or for any positive density subfamily thereof. For certain other families of elliptic curves over $\mathbb{Q}$, such as those with one or two marked points, we prove that the average of the number of integral points is bounded; in fact, for the family with one marked point, the $r$-th moment is also bounded for all positive $r\leq \log_2 3$. The essential new ingredient in our proof is an upper bound on the number of $S$-integral points on an affine integral Weierstrass model $\mathcal{E}$ of an elliptic curve $E$ over $K$ depending on the rank of the curve, the class group and degree of $K$, and the number of primes of $K$ whose square divides the discriminant of $\mathcal{E}$. For example, over $\mathbb{Q}$, the bound for integral points on $\mathcal{E}$ is $2^{\mathrm{rank}{E(\mathbb{Q})}} O(1)^s$, where $s$ is the number of prime squares dividing the discriminant of $\mathcal{E}$. The theorems on moments then follow from averaging this new upper bound; crucially, we can bound the average of the $2^{\mathrm{rank}}$ term by using results on the average sizes of Selmer groups in the families. In order to prove the bounds for the $r$-th moment when $r= \log_2 5$ (and the analogous equality cases for the other families), we introduce a method to count orbits of coregular representations with possibly unbounded weights.