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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 $X$?

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The answer is no even for gonality 2. Try to prove that. – QiL'8 Jan 31 at 16:31
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The curve $y^2=x^{2g+1}+1$ admits the automorphism $(x,y)\mapsto (\zeta_{2g+1}^n x, y)$ for all $n=1,\ldots,2g+1$. Thus, the number of automorphisms of this hyperelliptic curve is at least $2g+1$. Is this correct? – Harry Feb 2 at 9:34
Yes in characteristic $0$. Otherwise consider $y^2=x^{p^n}-x$ in characteristic $p>2$. – QiL'8 Feb 2 at 17:29

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