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This is a Quantitative Study study.

October 10, 2025Open Access

Elliptic curves and rational points in arithmetic progression

Key Points

The length of rational sequences with arithmetic progression is upper bounded by A^r, relating to the Mordell-Weil rank.
Assuming Lang's conjecture, sequences of rational points exist with well-defined upper limits based on curve rank.
The findings suggest uniform bounds on sequence lengths across different elliptic curves over rational numbers.
This work contributes to the understanding of rational points’ distribution on elliptic curves, with implications for Lang's conjecture.

Abstract

Let E/Q be an elliptic curve. We consider finite sequences of rational points \P₁, , PN\ whose x-coordinates form an arithmetic progression in Q. Under the assumption of Lang's conjecture on lower bounds for canonical height functions, we prove that the length N of such sequences satisfies the upper bound Aʳ, where A is an absolute constant and r is the Mordell-Weil rank of E/Q. Furthermore, assuming the uniform boundedness of ranks of elliptic curves over Q, the length N satisfies a uniform upper bound independent of E.

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Cite This Study

S. L. G. Choi (Sat,) studied this question.

synapsesocial.com/papers/68e865117ef2f04ca37e4e66 https://doi.org/https://doi.org/10.48550/arxiv.2510.03828

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