Boundedness of average rank of elliptic curves ordered by the coefficients

2025/06/08 by Fatemehzahra Janbazi, Janbazi, Fatemehzahra
Mathematics · Computer Science · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic

paper · pdf · doi:10.48550/arxiv.2506.07089

Abstract

We study the average rank of elliptic curves E_A,B : y² = x³ + Ax + B over ℚ, ordered by the height function h(E_A,B) := max(|A|, |B|). Understanding this average rank requires estimating the number of irreducible integral binary quartic forms under the action of GL₂(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.

Citations

• The second moment of the size of the 2-Selmer group of elliptic curves
• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• Binary quartic forms having bounded invariants, and the boundedness of\n the average rank of elliptic curves
• On the rank of elliptic curves
• The density of discriminants of quintic rings and fields

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• The average number of integral points on elliptic curves is bounded