Cubic Progressions on Smooth Plane Curves

Abstract

Let $C \subset \Pbb^2_k$ be a smooth projective plane curve of degree $d \ge 5$ 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 $C(k) \neq \emptyset$ admits infinite cubic progressions, so no analogue of the main theorem can hold there.