Riemann-Hurwitz for plane meromorphic / sphere maps

Intuition [Beginner]

A map between surfaces can fold. Think of pressing a sphere onto another sphere: most points have one preimage, but at fold points several sheets come together. The Riemann-Hurwitz formula counts how these folds change the shape of the surface.

The formula relates the "number of holes" (genus) of the domain surface to the number of holes of the target, the degree of the map, and a correction term counting the folds. If the map has degree $d$, each fold contributes a penalty that reduces the Euler characteristic.

The Euler characteristic of a surface counts vertices minus edges plus faces in any triangulation. For the sphere it is $2$; for a torus it is $0$; for a genus-$g$ surface it is $2-2g$. The Riemann-Hurwitz formula tells you exactly how the Euler characteristic changes across a branched cover.

Visual [Beginner]

Two spheres drawn side by side. An arrow between them represents a degree-3 map. On the target sphere, three points are marked as branch points. On the domain sphere, three points are marked where the map folds, each labelled with its ramification index. The formula $2=3\cdot 2-(2+2+2)$ is written below, confirming the Euler characteristic balance.

The picture shows that branch points act as "defects" in the covering. The more folds, the lower the Euler characteristic of the domain.

Worked example [Beginner]

Consider the map $f$ from the Riemann sphere to itself given by $f(z)=z^3$. This map has degree $3$: every point (except the two branch points) has exactly $3$ preimages.

Step 1. The branch points are $z=0$ and $z=\infty$. At $z=0$, the map wraps three sheets together, so the ramification index is $3$. The same holds at $\infty$.

Step 2. Compute the penalty. At each branch point, the contribution is $e_p-1=3-1=2$. The total penalty is $2+2=4$.

Step 3. The Euler characteristic of the sphere is $2$. The formula gives $\chi (\text{domain})=3\cdot 2-4=2$, which is correct since the domain is also a sphere.

What this tells us: when the domain and target are both spheres, the branch points exactly compensate for the degree, and the topology matches.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Ramification index). Let $f:X\to Y$ be a non-constant holomorphic map between Riemann surfaces and let $p\in X$. There exists a local coordinate $z$ near $p$ with $z(p)=0$ and a local coordinate $w$ near $f(p)$ with $w(f(p))=0$ such that $w\circ f=z^e$ for some integer $e\ge 1$. This integer $e=e_p$ is the ramification index of $f$ at $p$. The point $p$ is a ramification point if $e_p\ge 2$; its image $f(p)$ is a branch point.

Definition (Degree). A non-constant holomorphic map $f:X\to Y$ between compact Riemann surfaces has a well-defined degree $d=\mathrm{deg}f$: for every $q\in Y$ that is not a branch point, the fiber $f^{-1}(q)$ has exactly $d$ points.

Definition (Euler characteristic of a compact Riemann surface). The Euler characteristic of a compact Riemann surface $X$ of genus $g$ is $\chi (X)=2-2g$.

Counterexamples to common slips [Intermediate+]

• Confusing ramification points with branch points. The ramification point is in the domain; the branch point is its image in the target. A branch point can have several ramification points above it, each with different ramification indices.
• Assuming the ramification index is always $2$. For $f(z)=z^n$ at $z=0$, the ramification index is $n$. Higher ramification is common and important.
• Omitting ramification points over $\infty$. On the Riemann sphere, the point at infinity must be checked. The map $f(z)=z^3$ has ramification index $3$ at both $0$ and $\infty$.

Key theorem with proof [Intermediate+]

Theorem (Riemann-Hurwitz formula). Let $f:X\to Y$ be a non-constant holomorphic map of degree $d$ between compact Riemann surfaces. Then

$$\chi (X)=d\cdot \chi (Y)-\sum _{p\in X}(e_p-1).$$

Equivalently, if $Y$ has genus $g_Y$ and $X$ has genus $g_X$, then

$$2g_X-2=d(2g_Y-2)+\sum _{p\in X}(e_p-1).$$

Proof. Choose a triangulation of $Y$ that includes all branch points $q_1,\dots ,q_m$ as vertices. Subdivide if necessary so that each triangle lies in a single coordinate chart and each branch point is a vertex.

Lift the triangulation to $X$ via $f$. Over each non-branch-point vertex, there are exactly $d$ preimage vertices. Over a branch point $q_j$, the number of preimage vertices is $d-{\sum }_{p\in f^{-1}(q_j)}(e_p-1)$, because each ramification point of index $e_p$ identifies $e_p$ sheets into one.

Let $V_Y,E_Y,F_Y$ be the numbers of vertices, edges, and faces of the triangulation on $Y$. Then $\chi (Y)=V_Y-E_Y+F_Y$.

Lift the triangulation: over each face of $Y$, there are $d$ faces in $X$, so $F_X=d\cdot F_Y$. Over each edge of $Y$, there are $d$ edges in $X$ (since the map is locally biholomorphic away from ramification points, and no edge passes through a branch point), so $E_X=d\cdot E_Y$. For vertices, $V_X=d\cdot V_Y-{\sum }_{p\in X}(e_p-1)$, because at each branch point the $d$ sheets merge, with the deficit at each ramification point $p$ being $e_p-1$.

Compute:

$$\chi (X)=V_X-E_X+F_X=d\cdot V_Y-\sum (e_p-1)-d\cdot E_Y+d\cdot F_Y$$

$$=d(V_Y-E_Y+F_Y)-\sum (e_p-1)=d\cdot \chi (Y)-\sum _{p\in X}(e_p-1).\square$$

Bridge. This proof lifts a triangulation from the target to the domain, and the foundational reason the formula works is that branching creates a deficit of vertices relative to the expected $d$-sheeted cover. This pattern appears again in the Riemann-Roch theorem where the degree-genus relation gets refined by divisor data. The formula identifies the topological effect of branching with an algebraic invariant, and the bridge is that the Euler characteristic of the domain is entirely determined by the degree, the Euler characteristic of the target, and the ramification data. This generalises to maps between higher-dimensional manifolds via the Grothendieck-Riemann-Roch theorem.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Hurwitz's automorphism theorem). Let $X$ be a compact Riemann surface of genus $g\ge 2$. Then $|\mathrm{Aut}(X)|\le 84(g-1)$.

The proof applies Riemann-Hurwitz to the quotient map $X\to X/\mathrm{Aut}(X)$. The quotient is ${\mathbb{P}}^1$ (by the classification of compact Riemann surfaces), and the ramification data constrains the group order. The bound $84(g-1)$ is sharp: the Klein quartic ($g=3$, $168=84\cdot 2$ automorphisms) and the Bring curve ($g=4$, $120=84\cdot \frac{10}{7}$ automorphisms up to the bound) realise it.

Theorem 2 (Belyi's theorem, 1979). A compact Riemann surface $X$ can be defined over a number field (i.e., $X$ is the complex points of an algebraic curve defined over $\overline{\mathbb{Q}}$) if and only if there exists a holomorphic map $f:X\to {\mathbb{P}}^1$ branched over at most three points.

Belyi's theorem is remarkable: an arithmetic condition (definability over $\overline{\mathbb{Q}}$) is equivalent to a topological condition (existence of a three-point branched cover of the sphere). Grothendieck called this the "children's drawings" (dessins d'enfants) correspondence.

Theorem 3 (Riemann-Hurwitz for non-compact surfaces). The formula extends to non-compact surfaces via triangulations with infinitely many cells, provided the map is proper and the ramification sum converges. For a proper holomorphic map $f:X\to Y$ between possibly non-compact Riemann surfaces, ${\chi }_c(X)=d\cdot {\chi }_c(Y)-\sum (e_p-1)$ where ${\chi }_c$ denotes Euler characteristic with compact support.

Theorem 4 (Additivity of the degree). For a composition $g\circ f:X\to Z$, the ramification indices satisfy $e_p(g\circ f)=e_p(f)\cdot e_{f(p)}(g)$. The degree is multiplicative: $\mathrm{deg}(g\circ f)=\mathrm{deg}(f)\cdot \mathrm{deg}(g)$.

Theorem 5 (Existence of branched covers). Given a compact Riemann surface $Y$ of genus $h$ and a set of points $q_1,\dots ,q_m\in Y$ with partition data ${\lambda }_1,\dots ,{\lambda }_m$ of $d$, there exists a degree-$d$ cover $f:X\to Y$ with the specified branching at the $q_j$ if and only if the Riemann-Hurwitz formula gives a non-negative integer genus for $X$ and the monodromy representation generates a transitive subgroup of $S_d$.

Theorem 6 (Grothendieck-Riemann-Roch connection). The Riemann-Hurwitz formula is the genus-$0$ shadow of the Grothendieck-Riemann-Roch theorem applied to the structure sheaf. The higher-dimensional generalisation replaces the Euler characteristic with the Todd genus and the ramification sum with the Chern classes of the relative cotangent bundle.

Synthesis. The Riemann-Hurwitz formula is the foundational reason that the topology of a branched cover is entirely determined by local data at the branch points. The central insight is that the Euler characteristic, a global invariant, decomposes as a degree-weighted copy of the target's topology minus a local correction. This is exactly the pattern that appears again in the Grothendieck-Riemann-Roch theorem where the Chern character and Todd genus play the roles of the degree and Euler characteristic. Putting these together, the formula identifies the ramification divisor with the topological defect of the cover, and the bridge is that every question about the topology of holomorphic maps reduces to counting ramification points. This generalises from curves to higher dimensions via the Riemann-Roch formalism, and the pattern recurs in the theory of dessins d'enfants where Belyi's theorem converts arithmetic into combinatorics of branched covers.

Full proof set [Master]

Proposition 1 (Hurwitz's automorphism theorem). Let $X$ be a compact Riemann surface of genus $g\ge 2$ and $G=\mathrm{Aut}(X)$. Then $|G|\le 84(g-1)$.

Proof. The group $G$ acts holomorphically on $X$. The quotient $Y=X/G$ is a compact Riemann surface. The quotient map $\pi :X\to Y$ has degree $|G|$.

Let $Y$ have genus $h$. By Riemann-Hurwitz applied to $\pi$:

$$2g-2=|G|(2h-2)+\sum _{p\in X}(e_p-1).$$

The ramification points of $\pi$ lie over the branch points $q_1,\dots ,q_s$ of $Y$. For each $q_j$, the stabilizer of any point in ${\pi }^{-1}(q_j)$ is cyclic of order $e_j$. By the orbit-stabilizer theorem, ${\pi }^{-1}(q_j)$ consists of $|G|/e_j$ points, each with ramification index $e_j$. The contribution from $q_j$ to the sum is $(|G|/e_j)(e_j-1)=|G|(1-1/e_j)$.

So: $2g-2=|G|[(2h-2)+{\sum }_{j=1}^s(1-1/e_j)]$.

Let $B=(2h-2)+{\sum }_j(1-1/e_j)$. Since $g\ge 2$, we have $2g-2>0$, so $B>0$. The possible cases for $(h,s,e_j)$ are constrained by $B>0$ with $e_j\ge 2$:

• $h=0$, $s=3$: $B=-2+(1-1/e_1)+(1-1/e_2)+(1-1/e_3)=1-(1/e_1+1/e_2+1/e_3)$. The minimum positive value is $1/42$ (from $(e_1,e_2,e_3)=(2,3,7)$).
• $h=0$, $s\ge 4$: $B\ge 1/2$.
• $h\ge 1$: $B\ge 1$.

In the worst case $B=1/42$, giving $|G|=(2g-2)/B\le 84(g-1)$. $\square$

Proposition 2 (Composition multiplicativity of ramification). For holomorphic maps $f:X\to Y$ and $g:Y\to Z$ between compact Riemann surfaces and $p\in X$: $e_p(g\circ f)=e_p(f)\cdot e_{f(p)}(g)$.

Proof. Choose local coordinates $z$ at $p$, $w$ at $f(p)$, and $\zeta$ at $g(f(p))$ so that $w\circ f=z^{e_p(f)}$ and $\zeta \circ g=w^{e_{f(p)}(g)}$. Then $\zeta \circ (g\circ f)=(\zeta \circ g)\circ f=(w^{e_{f(p)}(g)})\circ f=(z^{e_p(f)}{)}^{e_{f(p)}(g)}=z^{e_p(f)\cdot e_{f(p)}(g)}$. By uniqueness of the ramification index in local coordinates, $e_p(g\circ f)=e_p(f)\cdot e_{f(p)}(g)$. $\square$

Connections [Master]

• Holomorphic functions and the argument principle 06.01.01. The ramification index at a point of a holomorphic map between Riemann surfaces generalises the winding number of a holomorphic function around a zero. The argument principle counts zeros with multiplicity; Riemann-Hurwitz counts the topological effect of those multiplicities in aggregate.

• The Riemann sphere 06.01.07. The Riemann sphere ${\mathbb{P}}^1$ is the target of the most important class of branched covers. Maps ${\mathbb{P}}^1\to {\mathbb{P}}^1$ given by rational functions $f(z)=p(z)/q(z)$ are the prototype to which Riemann-Hurwitz applies, and the formula determines the genus of more general surfaces realised as branched covers of the sphere.

• The modular function and j-invariant 06.01.26. The modular function $\lambda :\mathbb{H}\to \mathbb{C}\setminus \{0,1\}$ extends to a branched cover of the sphere, and the branching data at $0,1,\infty$ is computed by the Riemann-Hurwitz formula. This connects the modular function to the topology of the moduli space of elliptic curves.

Historical & philosophical context [Master]

Riemann introduced the genus and the branching relation in his 1857 paper on Abelian functions ^{[Riemann 1857]}, as part of his topological classification of surfaces. The explicit formula connecting genus, degree, and ramification was given by Riemann and then systematised by Hurwitz in 1891 ^{[Hurwitz 1891]}, who used it to study the existence and classification of Riemann surfaces with prescribed branching.

Hurwitz's automorphism theorem, proved in the same 1891 paper, remained the primary tool for studying automorphism groups of compact Riemann surfaces until the advent of algebraic geometry in the 20th century. Belyi's 1979 theorem ^{[Belyi 1979]} connected the Riemann-Hurwitz framework to arithmetic geometry, showing that branched covers of ${\mathbb{P}}^1$ with three branch points characterise algebraic curves defined over number fields.

Bibliography [Master]

@article{riemann1857,
  author = {Riemann, Bernhard},
  title = {Theorie der {A}bel'schen {F}unctionen},
  journal = {J. reine angew. Math.},
  volume = {54},
  pages = {115--155},
  year = {1857}
}

@article{hurwitz1891,
  author = {Hurwitz, Adolf},
  title = {{\"U}ber {R}iemann'sche {F}l{\"a}chen mit gegebenen {V}erzweigungspunkten},
  journal = {Math. Ann.},
  volume = {39},
  pages = {1--61},
  year = {1891}
}

@article{belyi1979,
  author = {Belyi, Gennadii},
  title = {On {G}alois extensions of a maximal cyclotomic field},
  journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
  volume = {43},
  pages = {267--276},
  year = {1979}
}

@book{miranda-curves,
  author = {Miranda, Rick},
  title = {Algebraic Curves and Riemann Surfaces},
  publisher = {AMS},
  year = {1995}
}

@book{farkas-kra,
  author = {Farkas, Hershel and Kra, Irwin},
  title = {Riemann Surfaces},
  publisher = {Springer},
  year = {1980}
}