[Paper Review] The average number of integral points on elliptic curves is bounded

Levent Alpöge|arXiv (Cornell University)|Dec 2, 2014

Algebraic Geometry and Number Theory23 references7 citations

TL;DR

This paper proves that the average number of integral points on elliptic curves over ℚ, ordered by height, is bounded above by 66. Using a combination of Bhargava-Shankar's methods, local height analysis, and a novel application of spherical codes and Roth-type estimates, the author establishes uniform bounds across families including Mordell curves and congruent number curves, resolving a long-standing open problem in arithmetic geometry.

ABSTRACT

We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By "$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal Weierstrass model $E_{A,B}: y^2 = x^3 + Ax + B$ with which one computes the na\"ıve height. The methods combine ideas from work of Silverman, Helfgott, and Helfgott-Venkatesh with work of Bhargava-Shankar and a careful analysis of local heights for "most" elliptic curves. The same methods work to bound integral points on average over the families $y^2 = x^3 + B$, $y^2 = x^3 + Ax$, and $y^2 = x^3 - D^2 x$.

Motivation & Objective

• To resolve the open question of whether the average number of integral points on elliptic curves over ℚ is bounded when ordered by height.
• To extend boundedness results to special families of elliptic curves, including $y^2 = x^3 + B$, $y^2 = x^3 + Ax$, and $y^2 = x^3 - D^2x$.
• To provide an effective upper bound on the average number of integral points, improving upon heuristic expectations.
• To unify techniques from Siegel's theorem, the Mumford gap principle, and modern Bhargava-Shankar methods to control integral points via local height and lattice structure.

Proposed method

• Uses Bhargava-Shankar's density results on ranks to classify curves by Mordell-Weil rank and control integral points via rank-dependent bounds.
• Applies a refined gap principle based on local heights to show that integral points cannot cluster too closely in the Mordell-Weil lattice.
• Decomposes the set of integral points into four classes (I–IV), proving classes I, II, and III are small via rational point distribution and spherical code bounds.
• Employs a bivariate Roth-type lemma to show class IV is empty for most curves, leveraging Diophantine approximation in two variables.
• Uses explicit optimization of spherical code bounds in $\mathbb{RP}^{r-1}$ for $3 \leq r \leq 13$, with numerical computation via Mathematica to minimize the upper bound.
• Restricts to the subfamily where $(A,B) \not\equiv (2,2) \pmod{3}$, where no integral points exist, and applies density estimates from Bhargava-Shankar and Bhargava-Skinner-Zhang to improve global bounds.

Experimental results

Research questions

• RQ1 Is the average number of integral points on elliptic curves over ℚ bounded when ordered by height?
• RQ2 Can the methods of Bhargava-Shankar and Helfgott-Venkatesh be extended to control integral points rather than just rational points?
• RQ3 Do families such as $y^2 = x^3 + B$, $y^2 = x^3 + Ax$, and $y^2 = x^3 - D^2x$ also have uniformly bounded average integral points?
• RQ4 Can spherical code bounds and local height analysis be combined to prove that integral points repel in the Mordell-Weil lattice?
• RQ5 What is the optimal effective upper bound on the average number of integral points across all elliptic curves?

Key findings

• The average number of integral points on all elliptic curves over ℚ, ordered by height, is bounded above by 66.
• Under the minimalist conjecture, the average number of integral points is at most $\frac{8}{9}$, significantly improving the unconditional bound.
• The bound extends to the families $y^2 = x^3 + B$, $y^2 = x^3 + Ax$, and $y^2 = x^3 - D^2x$, with the same upper bound of 66.
• For curves with $\mathrm{rank}(E) \leq 1$, the proportion is at least 84.22%, and the average number of integral points is bounded due to rank control.
• The methods yield a uniform bound of $\ll 1$ for the $k$-th moment of the number of integral points when $k \leq \log 5 / \log 3 \approx 1.4649$, implying rapid decay in the proportion of curves with many integral points.
• The final bound is optimized via numerical computation using spherical code bounds and a Mathematica implementation, with a choice of parameters $c = 0.998114$, $D = 612.117$, $s = 3$, and optimized $J$ for each $r$.