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Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$? Is it possible to do so if we know that $|G|$ is greater than the genus?

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  • $\begingroup$ Unlikely. There should be many function fields with zero automorphisms. However, large automorphism groups like $PGL(1)$ could uniquely identify the genus ($0$), like they do in the infinite field case. $\endgroup$
    – Will Sawin
    Feb 1, 2012 at 17:28
  • $\begingroup$ In case of finite fields there are many examples of large genus and large automorphism group like hermitian curve. $\endgroup$ Feb 1, 2012 at 19:22

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