Cubic Progressions on Smooth Plane Curves
Abstract
Let C ⊂ Pbb2k be a smooth projective plane curve of degree d ≥ 5 over a number field k. We prove that any cubic arithmetic or geometric progression sequence on C is finite. The proof rests on a prime-p tower of cyclic Kummer covers controlled by the divisor of a linear-form quotient on C, combined with the Kadets--Vogt classification of curves with infinitely many cubic points, the Abramovich--Harris gonality descent, a Castelnuovo--Severi factorization for the elliptic maps supplied by Kadets--Vogt independently at consecutive levels of the tower, and a deck-transformation obstruction. We also obtain a conditional quartic result under a cyclotomic non-splitting hypothesis. The case d = 4 with C(k) ≠ ∅ admits infinite cubic progressions, so no analogue of the main theorem can hold there.
Related
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BibTeX
@misc{badr2026cubic,
title = {{Cubic Progressions on Smooth Plane Curves}},
author = {Eslam Badr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04312},
eprint = {2610.04312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04312},
}
RIS
TY - UNPB
TI - Cubic Progressions on Smooth Plane Curves
AU - Eslam Badr
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04312
UR - https://arxiv.org/abs/2610.04312
AB - Let C ⊂ Pbb^2_k be a smooth projective plane curve of degree d ≥ 5 over a number field k. We prove that any cubic arithmetic or geometric progression sequence on C is finite. The proof rests on a prime-p tower of cyclic Kummer covers controlled by the divisor of a linear-form quotient on C, combined with the Kadets--Vogt classification of curves with infinitely many cubic points, the Abramovich--Harris gonality descent, a Castelnuovo--Severi factorization for the elliptic maps supplied by Kadets--Vogt independently at consecutive levels of the tower, and a deck-transformation obstruction. We also obtain a conditional quartic result under a cyclotomic non-splitting hypothesis. The case d = 4 with C(k) ≠ ∅ admits infinite cubic progressions, so no analogue of the main theorem can hold there.
ER -