The average number of integral points on the congruent number curves

Chan, Yik Tung

The average number of integral points on the congruent number curves

Chan S. 2024. The average number of integral points on the congruent number curves. Advances in Mathematics. 457(11), 109946.

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DOI

10.1016/j.aim.2024.109946

Journal Article | Published | English

Scopus indexed

Author

Chan, Stephanie^ISTA

Corresponding author has ISTA affiliation

Department

Browning Group

Abstract

We show that the total number of non-torsion integral points on the elliptic curves ED : y 2 = x3 − D2x, where D ranges over positive squarefree integers less than N, is O(N(log N) −1/4+ǫ). 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 group of the curves in this family.

Publishing Year

2024

Date Published

2024-11-01

Journal Title

Advances in Mathematics

Publisher

Elsevier

Volume

457

Issue

Article Number

109946

ISSN

0001-8708

eISSN

1090-2082

IST-REx-ID

18064

Cite this

Chan S. The average number of integral points on the congruent number curves. Advances in Mathematics. 2024;457(11). doi:10.1016/j.aim.2024.109946

Chan, S. (2024). The average number of integral points on the congruent number curves. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2024.109946

Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” Advances in Mathematics. Elsevier, 2024. https://doi.org/10.1016/j.aim.2024.109946.

S. Chan, “The average number of integral points on the congruent number curves,” Advances in Mathematics, vol. 457, no. 11. Elsevier, 2024.

Chan S. 2024. The average number of integral points on the congruent number curves. Advances in Mathematics. 457(11), 109946.

Chan, Stephanie. “The Average Number of Integral Points on the Congruent Number Curves.” Advances in Mathematics, vol. 457, no. 11, 109946, Elsevier, 2024, doi:10.1016/j.aim.2024.109946.

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