Examples of plane rational curves with two Galois points in positive characteristic, II

TL;DR

This paper constructs specific plane rational curves with two Galois points in positive characteristic fields, demonstrating the existence of curves with particular Galois groups at these points for certain degrees and characteristics.

Contribution

It provides explicit examples of plane rational curves with two Galois points having specified Galois groups in positive characteristic, expanding understanding of Galois points in algebraic geometry.

Findings

Existence of degree 12 curves with Galois group A4 in characteristic 11.

Existence of degree 24 curves with Galois group S4 in characteristic 23.

Existence of degree 30 curves with Galois group A5 in characteristic 59.

Abstract

It is proved that there exist plane rational curves of degree twelve (resp. twenty-four) with two different outer Galois points such that the Galois group at one of two Galois points is an alternating group $A_4$ (resp. a symmetric group $S_4$) of degree four, under the assumption that the characteristic of the ground field is eleven (resp. is twenty-three). For an alternating group $A_5$ of degree five, a similar existence theorem is confirmed, over a field of characteristic $59$, by GAP system.