Complete Determination of the Number of Galois Points for a Smooth Plane Curve

Abstract

Let C be a smooth plane curve. A point P in the projective plane is said to be Galois with respect to C if the function field extension induced by the projection from P is Galois. We denote by {\delta} (C) (resp. {\delta} '(C) ) the number of Galois points contained in C (resp. in {\mathbb P}^2 \setminus C ). In this article, we determine the numbers {\delta} (C) and {\delta} '(C) in any remaining open cases. Summarizing results obtained by now, we will present a complete classification theorem of smooth plane curves by the number {\delta} (C) or {\delta} '(C) . In particular, we give new characterizations of Fermat curve and Klein quartic curve by the number {\delta} '(C) .