List of Problems

(a) Find Galois points and the Galois groups for singular plane curves. – for smooth curves, the number of Galois points is at most three (resp. four) if they are outer (resp. inner). The Galois groups are cyclic. [46, 62] – (i) How is the structure of Galois group and how many Galois points do there exist? Is it true that the maximal number of outer (resp. inner) Galois points is three (resp. four)? – for rational curves [42, 68, 69, 72], for elliptic curves [34], for curves of prime degree [4] – (ii) Study the property of singularity when C has a Galois point. In particular, if a singular point is also Galois, how is the property of the singularity? Find the characterization of the curve with the maximal number of Galois points. – for lower degree or rational curves, [38, 39, 40, 56, 68, 69, 72] – (iii) Does there exist a curve with three Galois points such that their groups are not isomorphic to one another? More generally, consider the set