On the number of Galois points for a plane curve in positive characteristic, III

We consider the following problem: For a smooth plane curve C of degree d >= 4 in characteristic p > 0, determine the number delta(C) of inner Galois points with respect to C. This problem seems to be open in the case where d equivalent to 1 mod p and C is not a Fermat curve F(p(e) + 1) of degree p(e) + 1. When p not equal 2, we completely determine delta(C). If p = 2 (and C is in the open case), then we prove that delta(C) = 0,1 or d and delta(C) = d only if d - 1 is a power of 2, and give an example with delta(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of F(2)-rational points in P(2).