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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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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. –  Will Sawin Feb 1 '12 at 17:28
    
In case of finite fields there are many examples of large genus and large automorphism group like hermitian curve. –  Klim Efremenko Feb 1 '12 at 19:22
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