Abstract
We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p e + 1) of degree p e + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(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 \({\mathbb{F}_{2}}\) -rational points in \({\mathbb{P}^{2}}\).
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Fukasawa, S. On the number of Galois points for a plane curve in positive characteristic, III. Geom Dedicata 146, 9–20 (2010). https://doi.org/10.1007/s10711-009-9422-x
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DOI: https://doi.org/10.1007/s10711-009-9422-x