The ideal in $\mathbb{Z}[x]$ of all vanishing polynomials of a curve automorphism

Question

Let $C$ be a projective irreducible algebraic curve of genus $g$ over an algebraically closed field of characteristic $0$ (for simplicity). Given an automorphism $\alpha \in \mathrm{Aut}(C)$ of order $n$, we obviously have the equality $\alpha^n - 1 = 0$ in the endomorphism ring $\mathrm{End}(J)$ of the Jacobian $J$ of the curve $C$. Is there a way to find generators of the ideal $I_\alpha := \left\{ f \in \mathbb{Z}[x] \mid f(\alpha) = 0 \right\}$? For example, if $\alpha$ corresponds to an $n$-th primitive root of unity, then $I_\alpha = \langle \Phi_n \rangle$, where $\Phi_n$ is the $n$-th cyclotomic polynomial. I am especially interested in the situation when the Jacobian $J$ is completely decomposable into the $g$-th power of one elliptic curve $E$, that is $J \sim E^g$.

Answer 1

One way that should be pretty efficient is to first calculate the number of fixed points of $\alpha,\dots,\alpha^{n-1}$, then consider the function $g(i)$ which is the number of fixed points of $\alpha^i$ minus $2$ if $n\nmid i$ or $2g$ if $n\mid i$, then find lowest-degree solution $f(x) = \sum_{i=0}^{n-1} a_i x^i $ of the system of equations $\sum_{i=0}^{n-1} a_i g(i+j)=0$ for $j=0,\dots,n-1$ by linear algebra.

This works because, by the Lefschetz fixed point formula, $g(i)$ is the trace of $\alpha^i$ acting on $H^1$ of the curve (i.e. on $H^1$ of the Jacobian) and since $\alpha$ is semisimple, a linear combination of powers of $\alpha$ vanishes if and only if its trace when composed with any power of $\alpha$ vanishes.

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For $n$ large, a better approach may be to use the geometry of $C /\langle \alpha \rangle$ and the Euler characteristic formula for sheaves. As you point out, the only possible irreducible factors are cyclotomic polynomials for divisors $n'$ of $n$. Moreover:

If $C/\langle\alpha \rangle$ has genus $>1$ then the minimal polynomial is always $x^n-1$.

If $C / \langle \alpha \rangle$ has genus $1$ then $\phi_{n'}$ divides the minimal polynomial if and only if $n'=1$ or at least one point of $C/\langle \alpha \rangle$ has local monodromy of the covering $C \to C/\langle \alpha \rangle$ of order not dividing $n/n'$, i.e. if $ \alpha^{m}$ has a fixed point for $m$ not a multiple of $n'$.

If $C / \langle \alpha \rangle$ has genus $0$ then $\phi_{n'}$ divides the minimal polynomial if and only if at least three points of $C/\langle \alpha \rangle$ have order not dividing $n/n'$, i.e. if points in at least three different $\alpha$ orbits are fixed points of $\alpha^m$ for $m$ not a multiple of $n'$.

So in the genus $0$ and genus $1$ case one needs to find the fixed points of powers of $\alpha$, group them into orbits of $\alpha$, and note the least power fixing elements of each orbit. In the genus $1$ case you take $g$ the gcd of these powers and the minimal polynomial is $\frac{(x^n-1)(x-1)}{x^g-1} $ and in the genus $0$ case for each triple of orbits you take $\ell$ the lcm of these powers and then the minimal polynomial is $x^n-1$ divided by the gcd of the $x^\ell-1$s.