Certainly, if we fix the genus $g$ of a curve $X$, we have $\# $Aut$(X) \leq 84(g-1)$.
Let $X$ be a hyperelliptic curve. Is there a bound on $\#$Aut$(X)$? (Note that I do not want to fix the genus!)
More generally, can $\#$Aut$(X)$ be bounded in terms of the gonality of $X$?
via Recent Questions - Mathematics - Stack Exchange http://math.stackexchange.com/questions/291305/is-the-number-of-automorphisms-of-a-hyperelliptic-curve-bounded
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