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Every compact Riemann surface admits a nonconstant meromorphic function

Statement

Assume the full Axiom of Choice (The Axiom of Choice). Let $X$ be a compact Riemann surface of topological genus $g$ and let $p \in X$ . Put $D_{0} : = 2 g [p]$ and $F_{D_{0}} : = K_{X} \otimes O_{X} (- D_{0})$ , where $K_{X} : = Λ^{1, 0} T^{*} X$ is the canonical line bundle (Divisors, principal divisors and canonical divisors on a Riemann surface, Serre duality on a compact Riemann surface). Then:

$ℓ (D_{0}) = g + 1$ , and the intrinsic Riemann–Roch correction is $ℓ (D_{0}) - h^{0} (X, F_{D_{0}}) = g + 1.$ There exists a nonzero meromorphic differential $η$ on $X$ . For every such differential, put $K_{η} : = (η)$ ; then $\deg K_{η} = 2 g - 2$ , $ℓ (K_{η} - D_{0}) = 0$ , and the correction identity is $ℓ (D_{0}) - ℓ (K_{η} - D_{0}) = g + 1$ (Meromorphic differentials, orders and residues, The Riemann-Roch theorem on a compact Riemann surface). If $g \geq 1$ , a nonzero holomorphic differential exists as well.
There is a nonconstant meromorphic function whose only possible pole is $p$ , of order at most $2 g$ if $g \geq 1$ and at most $1$ if $g = 0$ . Viewed as a holomorphic map $X \to \hat{C}$ , it is proper and has degree at most $\max {1, 2 g}$ (Holomorphic maps and meromorphic functions on Riemann surfaces, Degree of a proper holomorphic map of Riemann surfaces).

Facts & Assumptions

Given: Full AC, a compact Riemann surface $X$ of topological genus $g$ , and a point $p \in X$ .

[F1]

Full AC is assumed by the cohomology, duality, and Riemann–Roch suppliers (The Axiom of Choice).

[F2]

For a divisor $D$ , $L (D)$ consists of $0$ and the meromorphic functions satisfying $(f) + D \geq 0$ ; at a point outside the support of $D$ , every element of $L (D)$ is holomorphic (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

For topological genus $g$ , $ℓ (0) = 1$ and $i (0) = h^{1} (X, O_{X}) = g$ (The Euler characteristic of the structure sheaf is one minus the genus).

[F4]

Serre duality identifies $i (0) = h^{1} (X, O_{X})$ with $h^{0} (X, K_{X})$ (Serre duality on a compact Riemann surface).

[F5]

For every divisor $D$ , $ℓ (D)$ and $i (D)$ are finite and the intrinsic formula is $ℓ (D) - h^{0} (X, K_{X} \otimes O_{X} (- D)) = \deg D + 1 - g$ (The Riemann-Roch theorem on a compact Riemann surface).

[F11]

A nonzero meromorphic differential exists, and for every such $η$ with $K_{η} = (η)$ , $i (D) = ℓ (K_{η} - D)$ (The Riemann-Roch theorem on a compact Riemann surface).

[F6]

A nonzero holomorphic section of $K_{X} = Λ^{1, 0} T^{*} X$ is a nonzero holomorphic, hence meromorphic, differential (Meromorphic differentials, orders and residues).

[F7]

A meromorphic function on $X$ is a holomorphic map $X \to \hat{C}$ other than the constant map at $\infty$ (Holomorphic maps and meromorphic functions on Riemann surfaces).

[F8]

A proper nonconstant holomorphic map between connected Riemann surfaces has positive degree $d = \sum_{x \in f^{- 1} (y)} e_{x} (f)$ , independent of $y$ (Degree of a proper holomorphic map of Riemann surfaces).

[F9]

On compact $X$ , a nonconstant meromorphic function is proper as a map to $\hat{C}$ (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F10]

If $X$ is compact and $\deg D < 0$ , then $L (D) = 0$ (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F12]

At a pole $q$ of a meromorphic function $f : X \to \hat{C}$ , its divisor order is $- e_{q} (f)$ (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F13]

Every principal divisor has degree zero (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F14]

All canonical divisors are linearly equivalent (Divisors, principal divisors and canonical divisors on a Riemann surface).

Proof

Riemann–Roch supplies a canonical divisor and computes its degree at every genus. The nonconstant function is obtained from $2 g [p]$ in positive genus and from $[p]$ in genus zero.

1.1F1F3F4F5F6F11F13F14givenchoosealgebra

By [F11], choose a nonzero meromorphic differential $η$ and put $K_{η} = (η)$ . By [F11] at $D = 0$ and [F3], $ℓ (K_{η}) = i (0) = g$ . At $D = K_{η}$ , [F11] gives $i (K_{η}) = ℓ (0) = 1$ , and [F5] gives $ℓ (K_{η}) - i (K_{η}) = \deg K_{η} + 1 - g$ . Thus $g - 1 = \deg K_{η} + 1 - g$ , so $\deg K_{η} = 2 g - 2$ for every genus. By [F13] and [F14], every other canonical divisor has this same degree. If $g \geq 1$ , [F3] and [F4] give $h^{0} (X, K_{X}) = g > 0$ , so [F6] also gives a nonzero holomorphic differential.

2.1F5F10F11step 1.1algebra

By [F5], the intrinsic correction at $D_{0}$ is $ℓ (D_{0}) - h^{0} (X, F_{D_{0}}) = 2 g + 1 - g = g + 1$ , and [F11] identifies $h^{0} (X, F_{D_{0}}) = i (D_{0}) = ℓ (K_{η} - D_{0})$ . By step 1.1, $\deg (K_{η} - D_{0}) = (2 g - 2) - 2 g = - 2$ at every genus, so [F10] gives $ℓ (K_{η} - D_{0}) = 0$ and hence $ℓ (D_{0}) = g + 1$ . In particular, when $g = 0$ , $D_{0} = 0$ and $ℓ (D_{0}) = 1$ ; no nonconstant function is inferred from this space.

3.1F1F2F3F5step 2.1algebra

If $g \geq 1$ , then $ℓ (D_{0}) = g + 1 \geq 2$ by step 2.1. The constants form a one-dimensional subspace of $L (D_{0})$ by [F3], so choose a nonconstant $f \in L (D_{0})$ . If $g = 0$ , [F5] applied to $[p]$ gives $ℓ ([p]) - i ([p]) = 2$ , hence $ℓ ([p]) \geq 2$ ; the constants again form a one-dimensional subspace, so choose a nonconstant $f \in L ([p])$ . By [F2], membership in these spaces means that all poles are confined to $p$ , with order at most $2 g$ in the first case and at most $1$ in the second.

4.1F7F8F9F12step 3.1∎

By [F7], $f$ is a nonconstant holomorphic map to $\hat{C}$ . It is proper by [F9], so [F8] computes its degree by the weighted fibre over $\infty$ . There are no poles away from $p$ , and at $p$ the local multiplicity equals the pole order by [F12]. Thus the degree is at most $2 g$ when $g \geq 1$ and at most $1$ when $g = 0$ ; in both cases it is at most $\max {1, 2 g}$ .

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Sources

Karl Otto Forster, Lectures on Riemann Surfaces (GTM 81, Springer 1981), translated by Bruce Gilligan (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)