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 SawinFeb 1, 2012 at 17:28
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$\begingroup$ In case of finite fields there are many examples of large genus and large automorphism group like hermitian curve. $\endgroup$– Klim EfremenkoFeb 1, 2012 at 19:22
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