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Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
Abstract
For integers t≥ −1, let θt be the largest real root of the Shanks polynomial gt(X)=X3−tX2−(t+3)X−1 and put Kt=ℚ(θt). We classify the points on En:y2=x3−n2x over Kt with abscissa θt−1 when n is a positive integer and En(ℚ) has rank zero. We determine the four pairs (d,t) with d>0 rational for which (θt−1)/d2 is an abscissa on E3(Kt). We also exclude the abscissa θt−1 on E5(Kt) and E6(Kt) and prove that only finitely many parameters t admit this abscissa for each fixed positive integer n.
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BibTeX
@misc{lu2026prescribed,
title = {{Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields}},
author = {Junyu Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08398},
eprint = {2609.08398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08398},
}
RIS
TY - UNPB
TI - Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
AU - Junyu Lu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.08398
UR - https://arxiv.org/abs/2609.08398
AB - For integers t≥ −1, let θ_t be the largest real root of the Shanks polynomial g_t(X)=X^3−tX^2−(t+3)X−1 and put K_t=ℚ(θ_t). We classify the points on E_n:y^2=x^3−n^2x over K_t with abscissa θ_t−1 when n is a positive integer and E_n(ℚ) has rank zero. We determine the four pairs (d,t) with d>0 rational for which (θ_t−1)/d^2 is an abscissa on E_3(K_t). We also exclude the abscissa θ_t−1 on E_5(K_t) and E_6(K_t) and prove that only finitely many parameters t admit this abscissa for each fixed positive integer n.
ER -