[Paper Review] On the number of quadratic twists with a rational point of almost minimal height

Joachim Petit|arXiv (Cornell University)|Apr 6, 2020

Analytic Number Theory Research28 references1 citations

TL;DR

This paper establishes an asymptotic formula for the number of quadratic twists of a fixed elliptic curve over Q that possess a rational point of almost minimal canonical height, analogous to Hooley's conjecture on fundamental units in real quadratic fields. By adapting techniques from analytic number theory and exploiting the geometry of 2-torsion points and line configurations on a Kummer surface, the author proves that the count of such twists grows like a power of log X, mirroring Hooley's result for number fields.

ABSTRACT

We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

Motivation & Objective

• . The paper aims to establish an analogue of Hooley's conjecture on fundamental units in real quadratic fields for the setting of elliptic curves.
• It investigates the frequency of quadratic twists of a fixed elliptic curve that have a rational point of almost minimal canonical height.
• The research seeks to quantify how often the generator of the Mordell–Weil group (modulo torsion) has height close to the theoretical minimum.
• The objective is to derive an asymptotic formula for the number of such twists, analogous to Hooley's asymptotic for the number of real quadratic fields with small fundamental unit.
• The study connects the arithmetic of elliptic curves to the geometry of Kummer surfaces and line configurations to achieve its result.

Proposed method

• . The author defines a counting function Nα(A, B; X) that counts square-free integers d ≤ X for which the minimal non-torsion canonical height on the quadratic twist Ed is at most d1/8+α.
• The method involves analyzing the geometry of the Kummer surface associated to the 2-torsion points of the elliptic curve E.
• It uses the group of isomorphisms of P1 preserving the 2-torsion points, Isom(P1; x(E[2])), to classify lines on the Kummer surface.
• The rational lines on the Kummer surface are determined by the rational 2-torsion points and the action of Galois on the 2-torsion structure.
• The proof relies on a detailed analysis of the automorphism group of the 2-torsion points and the conditions under which associated lines are rational.
• A key step is showing that only specific configurations of 2-torsion points and automorphisms yield rational lines, leading to a bound on the number of such twists.

Experimental results

Research questions

• RQ1 . What is the asymptotic number of quadratic twists of a fixed elliptic curve over Q that have a rational point of almost minimal canonical height?
• RQ2 . How does the distribution of such points compare to the distribution of small fundamental units in real quadratic fields, as studied by Hooley?
• RQ3 . What geometric and arithmetic conditions determine whether a line on the associated Kummer surface is rational?
• RQ4 . To what extent can the analogy between the unit group in number fields and the Mordell–Weil group in elliptic curves be made quantitative in the context of height bounds?
• RQ5 . What role do the 2-torsion points and their Galois action play in controlling the number of such twists?

Key findings

• . The paper establishes an asymptotic formula for the number of quadratic twists Ed with a rational point of height ≤ d1/8+α, analogous to Hooley's result for real quadratic fields.
• . The count Nα(A, B; X) grows asymptotically like X1/2(log X)2, matching the conjectured growth rate in Hooley's conjecture.
• . The main contribution is the proof of this asymptotic formula under the assumption that the elliptic curve has full 2-torsion rational over Q.
• . The proof relies on a precise count of rational lines on a Kummer surface associated to the 2-torsion structure of the curve.
• . It is shown that only specific configurations of 2-torsion points and automorphisms yield rational lines, which correspond to twists with small-height rational points.
• . The final bound Qα(X) ≪ǫ X13/32+13α/4+ǫ is shown to satisfy the required condition for Theorem 1 when α < 1/120, completing the proof.