Boundedness of average rank of elliptic curves ordered by the coefficients

TL;DR

This paper establishes an upper bound of 1.5 for the average rank of elliptic curves over rationals ordered by height, using advanced counting techniques for integral binary quartic forms and Selmer group analysis.

Contribution

It introduces a novel method for counting integral points in non-uniform regions, improving bounds on the average rank of elliptic curves.

Findings

Average rank of elliptic curves is bounded above by 1.5.

Develops a new counting technique for integral points in complex regions.

Provides bounds on the size of the 2-Selmer group.

Abstract

We study the average rank of elliptic curves $E_{A,B} : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$, ordered by the height function $h(E_{A,B}) := \text{max}(|A|, |B|)$. Understanding this average rank requires estimating the number of irreducible integral binary quartic forms under the action of $\mathrm{GL}_2(\mathbb{Z})$, where the invariants $I$ and $J$ are bounded by $X$. A key challenge in this estimation arises from working within regions of the quartic form space that expand non-uniformly, with volume and projection of the same order. To address this, we develop a new technique for counting integral points in these regions, refining existing methods and overcoming the limitations of Davenport's lemma. This leads to a bound on the average size of the 2-Selmer group, yielding an upper bound of 1.5 for the average rank of elliptic curves ordered by $h$.