Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its hyperelliptic involution?
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No, $\phi$ must come from an automorphism of $\mathbb{P}^1$ and that has fixed points.
answered Oct 9, 2012 at 18:18
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$\begingroup$ thank you for your comment. Can you give a little more detail? $\endgroup$– xuehangCommented Oct 10, 2012 at 2:10
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2$\begingroup$ Automorphisms of hyperelliptic curves (double covers of the projective line of genus bigger than one) must come from automorphisms of the projective line. This is a standard theorem. Now, an automorphism of the projective line is a linear fractional transformation and this always has fixed points (elementary exercise). Finally a fixed point on the projective line will lift to one of the two kinds of points in your question. $\endgroup$ Commented Oct 10, 2012 at 2:26