The average number of integral points on the congruent number curves

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown...

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Veröffentlicht in:Advances in mathematics (New York. 1965) 2024-11, Vol.457, p.109946, Article 109946
1. Verfasser: Chan, Stephanie
Format: Artikel
Sprache:eng
Schlagworte:
Elliptic curve
Integral point
Quadratic twist
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Zusammenfassung:We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.
ISSN:0001-8708
DOI:10.1016/j.aim.2024.109946