Let $\Lambda$ be a lattice in $\mathbb{C}$ and $X = \mathbb{C}/\Lambda$ be a complex torus. Exercise 6 of chapter 3 of Tamás Szamuely's book "Galois Groups and Fundamental Groups" (actually, the updated version in the erratum here) asks to show that $X$ have a Galois branched cover $Y \to X$, ramified over a single point of $x$, with Galois group $D_n$ (the dihedral group of order $2n$), and that such a cover must have $4$ branch points if $n$ is even, and $2$ if $n$ is odd.
I didn't have any trouble to show the existence of a Galois $D_n$-cover ramified over a single point of $X$, but I am at loss when it comes to show that the number of branch points is $4$ or $2$.
I am trying the odd case first. By considering the maximal cyclic subgroup of order $n$ of $D_n$, I can factor the cover $Y \to X$ to $Y \to Z \to X$, where $Z \to X$ is Galois, of group $\mathbb{Z}/2\mathbb{Z}$, and $Y \to Z$ is of group $\mathbb{Z}/n\mathbb{Z}$. Using Riemann-Hurwitz, I can show the cover $Y \to X$ is actually unramified, and that $Y$ is another complex torus, so that $Y \to Z$ is ramified over two points (which are the preimages in $Y$ of the point of $X$ at which $Z$ is ramified). I expect that the ramification of those points should be "nice", i.e that at those point have exactly one point in their preimage and that the ramification index (relative to $Y \to Z$) at these points of $Y$ is $n$, but I do not know how to show this. I tried further quotienting, but this is getting nowhere as it doesn't end up using the fact that the whole automorphism group is $D_n$.
Also, I feel like this approach has a flaw somewhere, since this would work as well in the even case (except if at some point in the proof of the fact that the points of $y$ have ramification $n$, the fact that $n$ is odd becomes essential), so I do not understand where the parity of $n$ should begin to be relevant in that approach.
For another approach, I know that there are $2$ characters $D_n \to \mathbb{C}^\times$ if $n$ is odd and $4$ such characters if $n$ is even, so I guess there has to be some connection with these characters. I guess the monodromy representation at each of those branch points give a character $D_n \to \mathbb{C}^\times$ but I don't know how to show that the branch point is uniquely determined by this character.
Any help or hint would be appreciated.
Edit: I had a look in Rick Miranda's "Algebraic curves and Riemann Surfaces", there is a chapter about monodromy, which contains a proposition that looks relevant to my problem. That is: there is an equivalence between Galois covers of degree $2n$ $X \setminus \{z_0\}$ and transitive actions of $\pi_1(X \setminus \{z_0\}, z)$ on a finite set of $2n$ elements (that part is the classical equivalence for covers), and the ramification indices can be computed by taking a small loop around $z_0$, and looking at the cycle decomposition of the permutation it induces.
But now, I am even more confused. There are two "obvious" actions of $D_{n}$ on a set with $2n$ elements: that is, its right/left action on itself, or its action by conjugation on itself. In both case, I have identified that the "small loop" around the removed point should actually correspond to the product of the two generating loops. Yet, when I use sage to compute the associated permutation of this representation, I get a composition of $n$ transpositions, regardless of parity, what goes wrong here? Is it my representation, the "small loop" around the removed point, or this approach as a whole?
It should be noted that in the book version of the exercise (not the erratum), the statement is to show there are $n$ branches points when $n$ is even, this is what I find in my computations above. Is the erratum version of the exercise even correct?
In a more concise manner, the question is the following: how many branch points does a Galois ramified cover of a complex torus with Galois group $D_n$ and ramified over a single point of the torus have, and how to show it?
I found an answer to my question and will post it here for closure.
The confusion about the number of branch points of the branched cover I constructed comes from the fact that I misindentified the "small loop" around the removed point of the torus. In the question, I wrote that this loop corresponds to the product of the two generators of the fundamental group of the torus minus point. But in fact, as one can see by identifying the torus with a square with glued edges, the "small loop" around a removed point is homotopic to the path going through the boundary of the square, which is the commutator of the two generators, and not their generators!
And now, everything goes as expected. The branched $D_n$-cover corresponds to the set $D_n$ with $D_n$ acting on it through the right action, and the commutator of the two generators of $D_n$ acts a product of two $n$-cycles if $n$ is odd, or the product of four $m$-cycles if $n = 2m$. The proposition from Miranda's book I mentionned in my first edit gives the result.