Descendants of algebraic curves admitting two Galois points

TL;DR

This paper explores the relationship between Galois points of algebraic curves and their quotients, introducing the concept of descendants for curves with two Galois points, and characterizes Fermat curves in this context.

Contribution

It introduces the notion of descendants of algebraic curves with two Galois points and proves that all descendants of Fermat curves are Fermat curves, characterizing when Fermat curves have descendants.

Findings

All descendants of Fermat curves are Fermat curves.

A Fermat curve has no descendant if and only if its degree is prime.

The paper establishes a connection between Galois points of curves and their quotients.

Abstract

A connection between Galois points of an algebraic curve and those of a quotient curve is presented; in particular, the notion of a descendant of algebraic curves admitting two Galois points is introduced. It is shown that all descendants of a Fermat curve are Fermat curves; in particular, a Fermat curve does not have a descendant if and only if the degree is a prime.