# On quadratic progression sequences on smooth plane curves

## TL;DR

This paper investigates the finiteness of quadratic progression sequences on smooth plane curves over number fields, linking their existence to the curve's gonality and hyperelliptic property.

## Contribution

It establishes finiteness of quadratic progressions on non-hyperelliptic curves and relates the infinitude of such progressions to the curve's arithmetic gonality.

## Key findings

- Quadratic progressions are finite on non-hyperelliptic curves.
- The arithmetic gonality determines the existence of infinite progressions.
- Hyperelliptic curves can have infinite quadratic progressions.

## Abstract

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$.