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Given a sufficiently nice perfect field $k$ and a smooth projective curve $C$ of genus $g_C>1$ over $k$, can one compute the automorphism group ${\rm Aut}(C)$? It is known that ${\rm Aut}(C)$ is always finite, e.g. of order at most $84(g-1)$ if ${\rm char}(k)=0$. The crucial case seems to be that of $k$ algebraically closed, and to determine the group it should be enough to know its order. So for concreteness let me ask:

Question: Is there an algorithm that takes $f\in\mathbb{Q}[X,Y]$ as input which defines a geometrically integral plane curve $f(X,Y)=0$ with smooth projective model $C$ of genus $g_C>1$, and outputs $n_f:=|{\rm Aut}(C_\mathbb{C})|$, the order of the automorphism group of $C$ over $\mathbb{C}$?

Remarks:

  • I am not asking for an efficient algorithm, just whether the function $f\mapsto n_f$ is computable. I expect that the answer is YES and that this is known, but I could not find a reference.
  • It seems that MAGMA can compute ${\rm Aut}(C)$ for curves of genus $g_C>1$ over certain base fields, which includes algebraically closed fields when the curve is hyperelliptic.
  • If I understand Theorem 30 in [1] correctly, when the plane curve $C: f=0$ is already smooth, all of its automorphisms are linear and so it is possible to count them. For the general case any such bound on the degrees should suffice.

[1] J. Kollár, Algebraic hypersurfaces, Bull. Amer. Math. Soc. 2019, https://www.ams.org/journals/bull/2019-56-04/S0273-0979-2019-01663-2/S0273-0979-2019-01663-2.pdf

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    $\begingroup$ For a smooth projectively normal curve of genus $g$ and degree $d$, every automorphism can be presented with a tuple of polynomials of degree $1+ \lceil \frac{g}{d} \rceil$ since by Riemann-Roch $\mathcal O( 1+ \lceil \frac{g}{d} \rceil) \otimes \sigma^* \mathcal O(-1)$ has a section and multiplying the pullback under $\sigma$ of the linear system corresponding to the embedding inside $H^0( C,\mathcal O(1))$ by such a section produces the desired tuple. If the curve is not projectively normal maybe you need a higher bound on the degree... $\endgroup$
    – Will Sawin
    Commented Apr 22 at 14:59

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F. Hess, An algorithm for computing isomorphisms of algebraic function fields, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 3076, Springer, Berlin, 2004, pp. 263–271,

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  • $\begingroup$ perfect, thank you! $\endgroup$
    – Arno Fehm
    Commented Apr 23 at 5:57

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