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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.

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thank you for your comment. Can you give a little more detail? –  xuehang Oct 10 '12 at 2:10
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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. –  Felipe Voloch Oct 10 '12 at 2:26

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