# All possible actions of a cyclic group on an algebraic curve

Written on May 24, 2012

I will present here a small program to compute (a superset of) all possible faithful actions of a cyclic group on a smooth algebraic curve (Riemann surface) of genus g, given the information on a part of the ramification. In other words, the program computes all possible cyclic subgroups of Aut(F) for some curve F of genus g. The actual possibilities may be fewer than the ones computed, because in this first version of the program the only ingredient is Riemann-Hurwitz formula, that is a necessary condition, but not sufficient.

The code is hosted on github.

Let $$F$$ be a smooth algebraic curve of genus $$g$$, and $$C_r$$ a cyclic group of order $$r \geq 2$$ acting faithfully on $$F$$. Let $$C = F/G$$ be the quotient curve, of genus $$h$$, and $$\pi\colon F \to C$$ the quotient morphism. Riemann-Hurwitz formula tells us that $2 g - 2 = r(2h - 2) + \sum_{p \in F} (|{\rm Stab}_{C_r}(p)| - 1).$

In other words, the “nice” formula $$2g-2 = r(2h-2)$$ must be amended with a term coming from the points in which the morphism $$\pi$$ is not étale. We can simplify the discrepancy using the fact that $$|{\rm Stab}_{C_r}(p)|$$ is the same for all points $$p$$ in the same orbit (that is, for all points $$p$$ over the same point $$q \in C$$), and we have a total contribution of $(|{\rm Stab}_{C_r}(p)| - 1) \cdot |C_r \cdot p| = \left( \frac{r}{|C_r \cdot p|} - 1\right) \cdot |C_r \cdot p| = r - |C_r \cdot p| = r - |\pi^{-1}(q)|$ from the points over $$q$$. Writing $$n_q = |\pi^{-1}(q)|$$, we can rewrite Riemann-Hurwitz as $2 g - 2 = r(2h - 2) + \sum_{q \in C} (r - n_q).$

The input data for the program are the genus $$g$$ of $$F$$ and the information on the counterimage on some branch points of $$C$$, that is, some points in which the counterimage fails to have $$r$$ points. For example, we may ask for all cyclic actions on a curve of genus $$4$$ such that there are two points with $$n_q = 1$$ and one point with $$n_q = 2$$.

The first observation is that $$r$$ is forced to be a multiple of the lcm of all prescribed $$n_q$$: if there are $$n_q$$ points in a orbit, an element of $$C_r$$ must have order exactly $$n_q$$, and so $$n_q \vert r$$.

The second observation is that we need to have at least $$g = 2$$ or $$g = 1$$ and one prescribed point of ramification to have a limit on $$r$$.

Let $$Q’$$ be the discrepancy in Riemann-Hurwitz formula. The third observation is that we can factor out the prescribed ramification as in $Q’ = Q + \sum_{q}(r - n_q) = Q + Nr - c,$ where $$q$$ varies among the points with prescribed ramification, $$N \geq 0$$ is the number of points of $$C$$ with prescribed ramification, $$c > 0$$ is a constant, and $$Q \geq 0$$ is the (potential) additional contribution from the ramification.

The program is now straightforward. We try every $$r$$ that is a multiple of the lcm, starting from $$2$$ upwards. For every such $$r$$ we can express the genus of $$C$$ as $h = \frac{(2-N)r + (2g - 2 + c) - Q}{2r} = \frac{(2-N)r + c’ - Q}{2r}.$ Since $$h$$ is a non-negative integer and the only parameter that can vary is the non-negative $$Q$$, there are only a finite number of values for $$h$$. For every such values, we need to find out if there is a configuration of branch points that can realize that contribution. To do so, we compute the possible contributions to the discrepancy for a branch point in $$C$$, that are of the form $$(i - 1) \cdot r / i$$ for a divisor $$i$$ of $$r$$, and we can recursively find out all linear combinations of those discrepancies that sum to $$Q$$.

The only thing still open is to find an upper limit for $$r$$. As before, solving for $$h$$ gives us $h = \frac{(2-N)r + c’ - Q}{2r}.$ Since $$h \geq 0$$ and $$r \geq 2$$, the numerator is non-negative. On the other hand, if we solve for $$Q$$ we obtain $Q = (2-N)r + c’ - 2 r h$ that is again non-negative.

We obtain three cases depending on the sign of the coefficient of $$r$$, that is, depending on the sign of $$2-N$$.

Case $$2-N < 0$$. In this case, we obtain $r \leq \mathrm{min} \left\{ \frac{c’ - 2 r h}{N-2}, \frac{c’ - Q}{N-2} \right\} \Longrightarrow r \leq \frac{c’}{N-2},$ since both $$h$$ and $$Q$$ are non-negative.

Case $$2-N = 0$$. Since $$Q \geq 0$$, we obtain that either $$h = 0$$ or $$r \leq c’/2h$$, so we need to care only for a bound when $$h = 0$$ (where, $$Q = c’$$ is constant). Note that for any $$r$$, the smallest non-zero contribution to $$Q$$ is $$(i-1)\cdot r/i$$ for the smallest non-trivial divisor $$i$$ of $$r$$. In particular, the smallest contribution is at least $$r/2$$. Therefore, we can use as a limit $$r \leq 2c’$$.

Case $$2-N > 0$$. Of course $$2-N \leq 2$$; we have $$Q = (2-N-2h)r + c’$$. If $$2-N-2h \leq 0$$ there are no problems since $$Q$$ is bounded and we can limit $$r$$ to twice the maximum $$Q$$ as in the previous case. If $$2-N-2h > 0$$, then $$Q = r + c’$$ or $$Q = 2r + c’$$, and it seems that $$r$$ cannot be bounded. What happens is that $$c’$$, the difference between $$Q$$ and the nearest multiple of $$r$$, become relatively smaller and smaller as $$r$$ increases, and then $$Q$$ cannot be realized with the possible contributions. It is hard enough to estimate a bound on $$r$$ based on this, but it is much easier to limit $$r$$ using a result of Wiman in 1895 that states that the maximum order of an automorphism of a Riemann surface is $$2(2g + 1)$$.