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Picard zero is the Jacobian

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact connected Riemann surface. Then the Abel-Jacobi homomorphism $u : {Div}^{0} (X) \to Jac (X)$ induces a canonical isomorphism of abelian groups ${Pic}^{0} (X) = {Div}^{0} (X) / ⁣ \sim \overset{≅}{\to} Jac (X),$ where ${Pic}^{0} (X)$ is the group of degree-zero divisor classes (The Picard group of divisor classes and its degree-zero part) and $Jac (X) = Ω (X)^{*} / Λ$ is the Jacobian (The Jacobian of a compact Riemann surface). The isomorphism is canonical: it depends only on $X$ and on the Abel-Jacobi construction, and in particular on no choice of base point, symplectic basis or basis of $Ω (X)$ . Equivalently, in the line-bundle reading, degree-zero holomorphic line bundles on $X$ are classified up to isomorphism by their Abel-Jacobi class in $Jac (X)$ .

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ , the Abel-Jacobi homomorphism $u$ , and the groups ${Div}^{0} (X)$ , $Prin (X)$ , ${Pic}^{0} (X)$ and $Jac (X)$ .

[F1]

$u : {Div}^{0} (X) \to Jac (X)$ is a group homomorphism, base-point free on degree-zero divisors (The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent).

[F2]

Kernel of $u$ equals the subgroup $Prin (X)$ of principal divisors (Abel's theorem for divisors, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

$u$ is surjective (Jacobi inversion).

[F4]

${Pic}^{0} (X) = {Div}^{0} (X) / Prin (X)$ is the quotient of the abelian group ${Div}^{0} (X)$ by its subgroup $Prin (X)$ ; its elements are the linear-equivalence classes $[D]$ (The Picard group of divisor classes and its degree-zero part, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F5]

First isomorphism theorem for groups: a homomorphism with kernel $N$ induces an isomorphism from the quotient by $N$ onto its image (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[F6]

The line-bundle dictionary of the Picard group identifies divisor classes with isomorphism classes of holomorphic line bundles, and the degree-zero part with degree-zero line bundles (The Picard group of divisor classes and its degree-zero part).

[F7]

Full AC is inherited from the Abel and inversion suppliers and from the meromorphic-section existence used in the line-bundle dictionary (The Axiom of Choice).

Proof

technique · direct

1.1F1F2F4F5

By [F1], $u : {Div}^{0} (X) \to Jac (X)$ is a homomorphism of abelian groups with kernel $\ker u = Prin (X)$ by [F2]. Since ${Pic}^{0} (X) = {Div}^{0} (X) / Prin (X)$ by [F4], the first isomorphism theorem [F5] gives an injective homomorphism $\overset{ˉ}{u} : {Pic}^{0} (X) \to Jac (X)$ , $[D] \mapsto u (D)$ .

2.1F3step 1.1

By [F3] the homomorphism $u$ is surjective, so $\overset{ˉ}{u}$ is surjective as well; hence $\overset{ˉ}{u}$ is an isomorphism of abelian groups.

2.2F1F4step 1.1

The isomorphism $\overset{ˉ}{u}$ is computed from $u$ alone, and $u$ and its divisor extension are defined from the intrinsic period pairing on $X$ ; by [F1] the values on ${Div}^{0} (X)$ do not depend on the chosen base point, and the definition of $Jac (X)$ as the quotient of $Ω (X)^{*}$ by the intrinsic lattice $Λ = e (H_{1} (X; Z))$ makes the isomorphism independent of the chosen symplectic basis or basis of $Ω (X)$ . Hence the isomorphism is canonical in the stated sense.

3.1F6step 2.1

Under the line-bundle dictionary [F6], the quotient ${Pic}^{0} (X)$ is the group of isomorphism classes of degree-zero holomorphic line bundles on $X$ , and the isomorphism $\overset{ˉ}{u}$ attaches to the class of such a bundle its Abel-Jacobi class $u (D)$ for any divisor $D$ with $O (D)$ the bundle.

4.1F7step 2.1step 2.2step 3.1∎

Steps 1.1-2.2 prove the existence and canonicity of the isomorphism and its line-bundle reading, under the inherited full AC of [F7].

Source notes

Forster's §§21.6-21.7 (Lectures on Riemann Surfaces, printed pp. 170-171) factor the Abel-Jacobi construction through ${Pic}^{0} (X)$ and prove injectivity by Abel's theorem and surjectivity by Jacobi inversion; Looijenga's Lemma 7.4 and Theorem 7.6 (printed pp. 60-63) and McMullen's Theorem 15.4 with Corollary 15.9 (printed pp. 129-130) state the same isomorphism. The item composes the two directions already proved in this batch through the first isomorphism theorem and records the canonicity.

Depends on

Used by

Period matrix and Jacobian of the pentagon curve Example
Periods of a complex torus Example
The Abel image in its Jacobian Example

Dependency tree · two levels

73 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karl Otto Forster, Lectures on Riemann Surfaces, GTM 81, 4th corrected printing (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)