Editing Hurwitz's automorphisms theorem (section)

== Statement and proof ==

'''Theorem''': Let <math>X</math> be a smooth connected Riemann surface of genus <math>g \ge 2</math>. Then its automorphism group <math>\operatorname{Aut}(X)</math> has size at most <math>84(g-1)</math>.

''Proof:'' Assume for now that <math>G = \operatorname{Aut}(X)</math> is finite (this will be proved at the end).
* Consider the quotient map <math>X \to X/G</math>. Since <math>G</math> acts by holomorphic functions, the quotient is locally of the form <math>z \to z^n</math> and the quotient <math>X/G</math> is a smooth Riemann surface. The quotient map <math>X \to X/G</math> is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non-trivial stabiliser. Let <math>g_0</math> be the genus of <math>X/G</math>.
* By the [[Riemann-Hurwitz formula]], <math display="block"> 2g-2 \ = \ |G| \cdot \left( 2g_0-2 + \sum_{i = 1}^k \left(1-\frac{1}{e_i}\right)\right) </math> where the sum is over the <math>k</math> ramification points <math>p_i \in X/G</math> for the quotient map <math> X \to X/G</math>. The ramification index <math>e_i</math> at <math>p_i</math> is just the order of the stabiliser group, since <math>e_i f_i = \deg(X/\, X/G)</math> where <math>f_i</math> the number of pre-images of <math>p_i</math> (the number of points in the orbit), and <math>\deg(X/\, X/G) = |G|</math>. By definition of ramification points, <math>e_i \ge 2</math> for all <math>k</math> ramification indices. 

Now call the righthand side <math>|G| R</math> and since <math>g \ge 2 </math> we must have <math>R>0</math>. Rearranging the equation we find:
* If <math>g_0 \ge 2</math> then <math>R \ge 2</math>, and <math>|G| \le (g-1) </math>
* If <math>g_0 = 1 </math>, then <math> k \ge 1</math>  and <math>R\ge 0 + 1 - 1/2 = 1/2</math> so that <math>|G| \le 4(g-1)</math>, 
* If <math>g_0 = 0</math>, then <math>k \ge 3</math> and
** if <math>k \ge 5</math> then <math>R \ge  -2 + k(1 - 1/2) \ge 1/2</math>, so that <math>|G| \le  4(g-1)</math>
** if <math>k=4</math> then <math> R \ge  -2 + 4 - 1/2  - 1/2  - 1/2 - 1/3 = 1/6</math>, so that <math>|G| \le 12(g-1)</math>,
** if <math>k=3</math> then write <math>e_1 = p,\, e_2 = q, \, e_3 = r</math>. We may assume <math>2 \le p\le q\ \le r</math>. 
*** if <math> p \ge 3 </math> then <math> R \ge  -2 + 3 - 1/3 - 1/3  - 1/4 = 1/12</math> so that <math>|G| \le 24(g-1)</math>,
*** if <math> p = 2 </math> then 
**** if <math>q \ge 4 </math> then <math>R \ge  -2 + 3 - 1/2 - 1/4  - 1/5 = 1/20</math> so that <math>|G| \le 40(g-1)</math>,
**** if <math>q = 3 </math> then <math>R \ge  -2 + 3 - 1/2 - 1/3  - 1/7 = 1/42</math> so that <math>|G| \le 84(g-1)</math>.
In conclusion, <math>|G| \le 84(g-1)</math>.

To show that <math>G</math> is finite, note that <math>G</math> acts on the [[cohomology]] <math>H^*(X,\mathbf{C})</math> preserving the [[Hodge decomposition]] and the [[Lattice (discrete subgroup)|lattice]] <math>H^1(X,\mathbf{Z})</math>.
*In particular, its action on <math>V=H^{0,1}(X,\mathbf{C})</math> gives a homomorphism <math>h: G \to \operatorname{GL}(V)</math> with [[discrete group|discrete]] image <math>h(G)</math>.
*In addition, the image <math>h(G)</math> preserves the natural non-degenerate [[Hilbert space|Hermitian inner product]] <math display="inline">(\omega,\eta)= i \int\bar{\omega}\wedge\eta</math> on <math>V</math>. In particular the image <math>h(G)</math> is contained in the [[unitary group]] <math>\operatorname{U}(V) \subset \operatorname{GL}(V)</math> which is [[Compact space|compact]]. Thus the image <math>h(G)</math> is not just discrete, but finite.  
* It remains to prove that <math>h: G \to \operatorname{GL}(V)</math> has finite kernel. In fact, we will prove <math>h</math> is injective. Assume <math>\varphi \in G</math> acts as the identity on <math>V</math>. If <math>\operatorname{fix}(\varphi)</math> is finite, then by the [[Lefschetz fixed-point theorem]], <math display="block"> |\operatorname{fix}(\varphi)| = 1 - 2\operatorname{tr}(h(\varphi)) + 1 = 2 - 2\operatorname{tr}(\mathrm{id}_V) = 2 - 2g < 0. </math>
This is a contradiction, and so <math>\operatorname{fix}(\varphi)</math> is infinite. Since <math>\operatorname{fix}(\varphi)</math> is a closed complex sub variety of positive dimension and <math>X</math> is a smooth connected curve (i.e. <math>\dim_{\mathbf C}(X) = 1</math>), we must have <math>\operatorname{fix}(\varphi) = X</math>. Thus <math>\varphi</math> is the identity, and we conclude that <math>h</math> is injective and <math>G \cong h(G)</math> is finite.
Q.E.D.

'''Corollary of the proof''': A Riemann surface <math>X</math> of genus <math>g \ge 2</math> has <math>84(g-1)</math> automorphisms if and only if <math>X</math> is a branched cover <math>X \to \mathbf{P}^1</math> with three ramification points, of indices ''2'',''3'' and ''7''.