Cubic Progressions on Smooth Plane Curves

Let $C\subset {\text{Pbb}}_k^2$ 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 $C(k)\mathrm{\ne }\mathrm{\varnothing }$ admits infinite cubic progressions, so no analogue of the main theorem can hold there.