Algebraic applications of Hurwitz' theorem

Question

Hurwitz' theorem states that for a finite separable morphism $f : X \to Y$ of curves of degree $n$ and with ramification divisor $R$, we have

$2 g(X) - 2 = n (2 g(Y) - 2) + \deg(R)$.

Besides, we have $\deg(R)=\sum_{p \in X} (e_p - 1)$ if $f$ has only tame ramification [Hartshorne, IV, Cor. 2.4]. One of the consequences is $g(X) \geq g(Y)$, which also holds when $f$ is not supposed to be separable [loc. cit. 2.5.4].

One purely algebraic application of Hurwitz' theorem is Luroth's theorem, which states every nontrivial intermediate field of $k(t)$ over $k$ is isomorphic to $k(x)$ over $k$. However, it is easy to give a direct algebraic proof of Luroth's theorem, even if $k$ is not supposed to be algebraically closed (which is probably needed for Hurwitz' theorem). Therefore I wonder if there are other algebraic application of Hurwitz' theorem using the correspondence between curves and function fields.

Question: Are there other interesting algebraic applications of Hurwitz' theorem?

Answer 1

As far as I remember the Riemann-Hurwitz-formula is used to prove the inequality

$|\mathrm{Aut}(F|K)|\leq 84(g-1)$

for the number of automorphisms of an algebraic function field $F$ of one variable over $K$, where $K$ has characteristic $0$ and $g\geq 2$ holds for the genus of $F|K$.

Answer 2

Notice that $\deg(R)=0$ and $g(Y)=1$ imply $g(X)=1$.

This means that every unramified cover of an curve of genus $1$ is again a curve of genus $1$. Translating into the algebraic language and using the fact that , if $\textrm{char}(k) \neq 2,3$, any curve of genus $1$ has a birational model of the form $y^2=x^3+px+q$, with $4p^3+27q^2 \neq 0$, we obtain the following result:

Assume $\textrm{char}(k)\neq 2,3$ and let $p, q \in k$ with $4p^3+27q^2 \neq 0$. Then every finite, unramified extension of the quotient field of

$k[x,y]/(y^2-x^3-px-q)$

can be written as the quotient field of

$k[s,t]/(s^2-t^3-at-b)$,

for some $a,b \in k$ with $4a^3+27b^2 \neq 0$.

Answer 3

On this other thread, about elliptic curves about function fields, the functional analogue of Mazur's theorem was evoked, and described as a simple consequence that the genus of modular curves grows with the conductor. This is indeed an application of the inequality $g(X)\geq g(Y)$ you mention.

Answer 4

Ree gave an application of Riemann-Hurwitz to permutation groups. If $g \in S_n$ (the symmetric group), then let $v(g)$ be $n$ minus the number of cycles of $G$. Ree observed that if $g_1,...,g_k \in S_n$ with the subgroup generated by them acting transitively on the set of $n$ elements, and with $v(g_1) + \cdots + v(g_k) = n-1$, then any product $g_{i_1} \cdots g_{i_k}$ is an $n$-cycle (where $i_j$ are the result of permuting the indices).

To prove this we form a branched cover over the sphere, branched over $k+1$ points. The first $k$ points we call $z_j$ and each has monodromy $g_{i_j}$. The last point $z_*$ has monodromy $(g_{i_1} \cdots g_{i_k})^{-1}$. Applying Riemann-Hurwitz to the resulting covering yields $$ v(g_1) + \cdots + v(g_k) + v((g_{i_1} \cdots g_{i_k})^{-1}) - 2n - 2 = 2g \geq 0 $$ Therefore, there is only one cycle in $(g_{i_1} \cdots g_{i_k})^{-1}$ and so there is only one cycle in $g_{i_1} \cdots g_{i_k}$. Ree mentions that he could not find a purely algebraic proof. I don't know if there is still no algebraic proof known.