The Riemann-Hurwitz Formula

I’ll basically be stealing from my other blog at http://categorymath.wordpress.com for the next three posts, since I really want to post but I am extremely busy. This post can be found here.

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 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 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 (recall c is as above). 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.

Best,

S.D.

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