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Principal divisor tests via the Abel-Jacobi map

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Sphere. For $X = \hat{C}$ one has $g = 0$ , $Ω (X) = 0$ and $Jac (X) = 0$ (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity, The space of holomorphic differentials and the degree of the canonical divisor, The Jacobian of a compact Riemann surface). Every degree-zero divisor $D = \sum_{a} n_{a} (a)$ on $\hat{C}$ is principal: the rational function $\prod_{a \in C, n_{a} \neq 0} (z - a)^{n_{a}}$ has divisor $D$ , since its order at infinity is $- \sum_{a \in C} n_{a} = n_{\infty}$ . Hence the Abel-Jacobi criterion of Abel's theorem for divisors is satisfied by all of them, and $u (D) = 0$ for every $D \in {Div}^{0} (\hat{C})$ . Explicitly, $((a) - (b)) = div (\frac{z - a}{z - b})$ for $a, b \in C$ .
Torus, one point. Let $X = C / Λ$ be a complex torus and $p, q \in X$ (Periods of a complex torus). Then $u ((q) - (p)) = q - p$ in the identification $Jac (X) = X = C / Λ$ ; hence $(q) - (p)$ is principal if and only if $q = p$ , because a meromorphic function on a torus with a single simple pole and zero would be a degree-one map to the sphere, which is impossible for genus one (A degree-one holomorphic map of compact Riemann surfaces is an isomorphism, Degree of a proper holomorphic map of Riemann surfaces).
Torus, symmetric pairs. For representatives $p, q \in C$ of points of $X$ , with $2 p, 2 q \notin Λ$ and $q \equiv̸ \pm p (m o d Λ)$ , the function $F (z) = \frac{℘ (z) - ℘ (q)}{℘ (z) - ℘ (p)}$ is a nonconstant meromorphic function on the torus with divisor $(q) + (- q) - (p) - (- p)$ (Weierstrass p function, Normal convergence, parity and periodicity of the Weierstrass p function, Divisor and residue laws for elliptic functions); correspondingly $u ((q) + (- q) - (p) - (- p)) = 0$ , since $q + (- q) - p - (- p) = 0$ in the group $C / Λ$ .
Non-principal examples. On a torus, any divisor of the form $(q) - (p)$ with distinct points $p, q \in X$ has $u \neq 0$ , so it is not principal. Equivalently, any plane representatives $\tilde{p}, \tilde{q} \in C$ satisfy $\tilde{q} - \tilde{p} \notin Λ$ . On a curve of genus $g \geq 2$ and for distinct points $p \neq q$ the divisor $(q) - (p)$ has $u \neq 0$ : a vanishing class would make $(q) - (p)$ principal by the criterion, hence produce a holomorphic map $X \to \hat{C}$ of degree one and an isomorphism $X ≅ \hat{C}$ , contradicting genus $g \geq 2$ (A degree-one holomorphic map of compact Riemann surfaces is an isomorphism).

Facts & Assumptions

Given: Full AC, the Riemann sphere, a complex torus $X = C / Λ$ , and a curve of genus $g \geq 2$ ; the Abel-Jacobi map $u$ in each case.

[F1]

The Abel-Jacobi criterion: for $D \in {Div}^{0} (X)$ , $D$ is principal if and only if $u (D) = 0$ (Abel's theorem for divisors, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F2]

On the sphere $g = 0$ and $Ω (X) = 0$ , so $Λ = 0$ and $Jac (X)$ is a point; meromorphic functions on the sphere are rational, and the divisor of $z \mapsto \frac{z - a}{z - b}$ is $(a) - (b)$ (The space of holomorphic differentials and the degree of the canonical divisor, The Jacobian of a compact Riemann surface, Meromorphic functions on the Riemann sphere are exactly the rational functions, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

On a complex torus $X = C / Λ$ the Abel-Jacobi map identifies $Jac (X)$ with $X$ and sends $(q) - (p)$ to $q - p$ in $X$ . This is zero exactly when $q = p$ , equivalently when any plane representatives satisfy $\tilde{q} - \tilde{p} \in Λ$ (Periods of a complex torus, The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent).

[F4]

A principal divisor $(q) - (p)$ with $p \neq q$ is the divisor of a meromorphic function whose associated map $X \to \hat{C}$ has degree one; a degree-one holomorphic map between compact connected Riemann surfaces is a biholomorphism (Divisors, principal divisors and canonical divisors on a Riemann surface, Degree of a proper holomorphic map of Riemann surfaces, A degree-one holomorphic map of compact Riemann surfaces is an isomorphism).

[F5]

The Weierstrass $℘$ -function is meromorphic, $Λ$ -periodic and even, with its only pole on the torus a double pole at $0$ (Weierstrass p function, Normal convergence, parity and periodicity of the Weierstrass p function). Subtracting a finite constant preserves that pole, so the descended function has degree two and total zero order two by the weighted fibre formula (Degree of a proper holomorphic map of Riemann surfaces, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F6]

$X = C / Λ$ is a compact connected Riemann surface of genus $1$ , and a curve of genus $g \geq 2$ is not isomorphic to the sphere (Periods of a complex torus, Genus and Euler characteristic of a compact Riemann surface).

[F7]

Full AC is inherited from the Abel-Jacobi construction (The Axiom of Choice).

Verification

Given: The objects and conventions in the Statement.

1.1F1F2

By [F2], $g (\hat{C}) = 0$ , $Ω (\hat{C}) = 0$ , $Λ = 0$ and $Jac (\hat{C}) = 0$ , so $u$ is the zero map; for a degree-zero divisor $D = \sum_{a} n_{a} (a)$ , set $f (z) = \prod_{a \in C, n_{a} \neq 0} (z - a)^{n_{a}}$ . At each finite $a$ its order is $n_{a}$ , and at infinity its order is $- \sum_{a \in C} n_{a} = n_{\infty}$ . Thus $(f) = D$ , so every such divisor is principal and satisfies the criterion [F1]; explicitly $((a) - (b)) = div \frac{z - a}{z - b}$ . This is claim 1.

1.2F3F4F6

On a torus, [F3] gives $u ((q) - (p)) = q - p$ , so the class vanishes exactly when $q = p$ ; if $q \neq p$ and $(q) - (p)$ were principal, then by [F4] there would be a degree-one map $X \to \hat{C}$ , hence $X ≅ \hat{C}$ of genus $0$ , contradicting the genus-one statement [F6]. This is claim 2.

1.3F1F3F5

For claim 3, evenness in [F5] makes $\pm q$ zeros of $℘ (z) - ℘ (q)$ ; they are distinct modulo $Λ$ because $2 q \notin Λ$ . The only pole is the double pole at $0$ , so the zero count in [F5] is two and these zeros are simple, with no others. The same argument applies to $p$ ; both numerator and denominator are $Λ$ -periodic, so the quotient $F$ descends to a meromorphic function on the torus with divisor $(q) + (- q) - (p) - (- p)$ (the double poles at $0$ cancel, while the simple zeros and poles at $\pm q, \pm p$ are disjoint because $q \equiv̸ \pm p$ and none is $0$ modulo $Λ$ ). Hence this divisor is principal and $u$ of it vanishes by [F1]; its group sum $q + (- q) - p - (- p) = 0$ is consistent with the torus identification [F3].

1.4F1F3F4F6

For claim 4, on a torus the class of $(q) - (p)$ is $q - p$ in $X$ by [F3], nonzero exactly when $q \neq p$ , equivalently $\tilde{q} - \tilde{p} \notin Λ$ for plane representatives. Such a divisor is not principal by [F1]. On a curve of genus $g \geq 2$ , if $p \neq q$ and $u ((q) - (p)) = 0$ , then [F1] makes $(q) - (p)$ principal, and [F4] yields a degree-one map $X \to \hat{C}$ and an isomorphism $X ≅ \hat{C}$ , contradicting $g \geq 2$ by [F6]. This is claim 4.

2.1F7step 1.1step 1.2step 1.3step 1.4∎

Claims 1-4 are steps 1.1-1.4, under the inherited AC of [F7].

Source notes

Forster's §20.8 (Lectures on Riemann Surfaces, printed pp. 165-166) states the Abel condition for doubly periodic functions, $\sum a_{k} \equiv \sum b_{k} (m o d Γ)$ , and the sphere and torus cases; McMullen (printed pp. 128-130) gives the same examples through $φ$ . The explicit divisor of $F$ in claim 3 is the standard $℘$ -computation, proved here through the zero/pole count of elliptic functions.

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Sources

Karl Otto Forster, Lectures on Riemann Surfaces, GTM 81, 4th corrected printing (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)