## The Riemann-Hurwitz Formula

This post is going to be one of those “occasional posts” that I talked about in the About page, which, as the reader has seen, is about the Riemann-Hurwitz formula.

I had already learnt about this formula a few years ago, and again a few weeks ago while perusing Hartshorne’s Algebraic Geometry, and came across it yet again while reading (or should I say glancing through) Diamond and Shurman’s A First Course in Modular Forms (which I call [DS] from now).love category theory, and am not an expert at number theory – this book seemed like an interesting book, and so I borrowed it from my local library. (The only parts I fully understood were the sections on topology, charts, and Riemann surfaces.)

Now to actual math. In [DS], section 3.1, one has the Riemann-Hurwitz formula. I’m going to try to explain this from a number-theorist point of view, so please feel free to point out any mistakes.

Let $X$ and $Y$ be two compact Riemann surfaces, with a nonconstant holomorphic map $f:X\to Y$. The first theorem is:

Theorem: $f:X\to Y$ is surjective.

Proof: The image $f(X)$ must be closed and open, thus $Y-f(X)$ is open, since compact sets are closed in Hausdorff spaces. Thus, $Y$ is disconnected, hence a contradiction. Q.E.D. $f$ has a number associated to it, known as the degree, a number $c\in \mathbb{Z}^+$ such that $|f^{-1}(y)|=d$ for all but finitely many $y\in Y$, which is defined by the following construction. Let $e_x$ denote the “multiplicity with which $f$ takes $0$ to $0$ as a map in local coordinates, making $f$ an $latex e_x$-to- $1$ map about $x$” ([DS]). Then, the degree is the positive integer such that $\sum_{x\in f^{-1}(y)}e_x=c$ for all $y\in Y$. (For a proof of the existence of $c$, see [DS, section 3.1].)

Let $g_X$ and $g_Y$ denote the genera of $latex X$ and $Y$ respectively (“genera” sounds weird, at least to me – but so does “genuses” – I guess “genii” would make sense, but that sounds a lot like the plural of “genie”…). The Riemann-Hurwitz formula is as follows:

Theorem (Riemann-Hurwitz Formula): $2g_X-2=\sum_{x\in X}(e_x-1)+c(2g_Y-2)$

Proof Sketch: There’s a very interesting proof (sketch) of this theorem, as [DS] gives. First, define $\mathcal{C}:=\{x\in X| e_x>1\}$. Triangulate $Y$ using $V_Y$ vertices including all points of $f(\mathcal{C})$, $F_Y$ faces and $E_Y$ edges. Under $f^{-1}$, we may lift this to a triangulation of $X$ with $E_X=cE_Y$ edges, $F_X=cF_Y$ faces, and by ramification, $V_X=cV_Y-\sum_{x\in X}(e_x-1)$ vertices. Since $2-2g_X=F_X-E_X+V_X$ and $2-2g_Y=F_Y-E_Y+V_Y$, the Riemann-Hurwitz formula follows.

A very detailed proof, which is in essence what I did when I expanded out the proof sketch in [DS], is here.