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Periods of a complex torus

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Λ = Z ω_{1} + Z ω_{2} \subseteq C$ be a full lattice with oriented basis and let $X = C / Λ$ be the associated complex torus with quotient map $q : C \to X$ (Complex lattice and quotient torus, The quotient $C / Λ$ is a compact Riemann surface). Then:

The translation-invariant differential $d z$ descends to a nowhere-vanishing holomorphic differential on $X$ ; $Ω (X) = C \cdot d z$ , so $X$ has genus $1$ , $ℓ (K) = 1$ and $\deg K = 0$ (The space of holomorphic differentials and the degree of the canonical divisor, Elliptic function for a lattice).
With $π_{1}, π_{2}$ the loops $t \mapsto [t ω_{1}]$ , $t \mapsto [t ω_{2}]$ , the periods of $d z$ are $P (π_{1}, d z) = ω_{1}$ and $P (π_{2}, d z) = ω_{2}$ (The period pairing and the period subgroup, Path integral of a holomorphic differential on a Riemann surface). Under the isomorphism $Ω (X)^{*} ≅ C$ , $α \mapsto α (d z)$ , the period lattice is exactly $Λ$ , and $Jac (X) = C / Λ = X .$
For every base point $p_{0} \in X$ the point Abel-Jacobi map $u_{p_{0}} : X \to Jac (X) = X$ is a biholomorphism, and for $D \in {Div}^{0} (X)$ one has $u (D) = 0$ if and only if $D$ is principal; equivalently ${Pic}^{0} (X) ≅ X$ canonically (The Abel-Jacobi map embeds a positive-genus surface, Picard zero is the Jacobian). In particular, for points $p, q \in X$ the class of the divisor $(q) - (p)$ is the class of $q - p$ in $C / Λ$ .

Facts & Assumptions

Given: Full AC, a full lattice $Λ = Z ω_{1} + Z ω_{2}$ , the torus $X = C / Λ$ , and the loops $π_{1}, π_{2}$ .

[F1]

$X$ is a compact Riemann surface, the quotient map $q$ is a holomorphic covering, and the charts are local inverses of $q$ with translation transitions (The quotient $C / Λ$ is a compact Riemann surface, Complex lattice and quotient torus).

[F2]

A holomorphic differential on $X$ is equivalently a $Λ$ -invariant holomorphic differential $h (z) d z$ on $C$ ; the differential $d z$ is invariant and nowhere vanishing, hence descends to a nowhere-vanishing holomorphic differential on $X$ (Meromorphic differentials, orders and residues, Elliptic function for a lattice).

[F3]

If $γ$ is a closed loop in $X$ and $\tilde{γ} : [0, 1] \to C$ is a lift, then $\int_{γ} d z = \tilde{γ} (1) - \tilde{γ} (0) \in Λ$ : the integral of $d z$ along a path is the difference of the endpoint values of any lift, because $z$ is a primitive of $d z$ on $C$ and $Λ$ is the group of deck translations. Conversely, for $λ \in Λ$ the projection of $t \mapsto t λ$ is a loop with $\int d z = λ$ . Hence the period subgroup $e (H_{1} (X; Z))$ equals $Λ$ (Complex lattice and quotient torus, The quotient $C / Λ$ is a compact Riemann surface, Path integral of a holomorphic differential on a Riemann surface).

[F4]

For a compact connected Riemann surface of genus $g$ , $\dim_{C} Ω (X) = g$ and a nonzero holomorphic differential has exactly $2 g - 2$ zeros counted with multiplicity (The space of holomorphic differentials and the degree of the canonical divisor).

[F5]

The path integral of a holomorphic differential is computed by local primitives; for $d z$ on $C$ a primitive is $z$ , so the integral along a lifted path is the difference of its endpoints; the period pairing $P$ agrees with integration over cycles and is additive (Path integral of a holomorphic differential on a Riemann surface, The period pairing and the period subgroup, The period pairing is well defined and computed by integration).

[F6]

The Jacobian is $Ω (X)^{*} / Λ^{'}$ with $Λ^{'} = e (H_{1} (X; Z))$ , the Abel-Jacobi map is represented by path integrals modulo $Λ^{'}$ , and for $g = 1$ it is a biholomorphism onto the Jacobian; ${Pic}^{0} (X) ≅ Jac (X)$ canonically (The Jacobian of a compact Riemann surface, The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent, The Abel-Jacobi map embeds a positive-genus surface, Picard zero is the Jacobian).

[F7]

Full AC is inherited from the Jacobian and classification interfaces; the example selects only the given lattice basis (The Axiom of Choice).

Verification

Given: The lattice, the torus and the two loops.

1.1F2F4

The translation action of $Λ$ on $C$ leaves $d z$ invariant, so by [F2] $d z$ descends to a holomorphic differential on $X$ , and it is nowhere vanishing because its local expressions are the constant coefficient $1$ . Hence $Ω (X) \neq 0$ , so $g \geq 1$ by [F4], and since a nonzero holomorphic differential has exactly $2 g - 2$ zeros counted with multiplicity while $d z$ has none, $2 g - 2 = 0$ and $g = 1$ ; then [F4] gives $\dim_{C} Ω (X) = 1$ and, since $d z \neq 0$ , $Ω (X) = C \cdot d z$ with $(d z) = 0$ of degree $0$ ; in particular $ℓ (K) = 1$ and $\deg K = 0$ .

1.2F3F5

The loops $π_{1}, π_{2}$ lift to the paths $t \mapsto t ω_{1}$ and $t \mapsto t ω_{2}$ on $[0, 1]$ ; by [F5] the path integral of $d z$ along $π_{1}$ is $ω_{1} - 0 = ω_{1}$ and along $π_{2}$ is $ω_{2}$ . By [F3] the period lattice is exactly $e (H_{1} (X; Z)) = Λ$ ; under $Ω (X)^{*} ≅ C$ , $α \mapsto α (d z)$ , the periods $ω_{1}, ω_{2}$ of the two standard loops are therefore the two generators of $Λ$ . Hence $Jac (X) = C / Λ = X$ .

2.1F1F5F6step 1.1step 1.2

By [F5] the point map sends $p = [z]$ to the class of the functional $ω \mapsto \int_{p_{0}}^{p} ω$ ; under the identification $Ω (X) = C d z$ of step 1.1 and $Jac (X) = C / Λ$ of step 1.2 this is the class of $z - z_{0}$ , that is, $u_{p_{0}} (p) = p - p_{0}$ in the group $X$ . Hence $u_{p_{0}}$ is the translation by $- p_{0}$ , a biholomorphism. Consequently $u (D) = 0$ for $D \in {Div}^{0} (X)$ exactly when $\sum_{p} n_{p} (p - p_{0}) = 0$ in $C / Λ$ , i.e. when the group sum of $D$ vanishes; by [F6] (Abel's criterion) this is exactly the condition that $D$ is principal, and ${Pic}^{0} (X) ≅ X$ . For $D = (q) - (p)$ the class is $q - p$ .

3.1F7step 1.1step 1.2step 2.1∎

The four displayed claims are steps 1.1, 1.2 and 2.1, under the inherited AC of [F7].

Source notes

Looijenga's Corollary 7.7 (Riemann Surfaces, printed p. 61) states that for genus one the point map is an isomorphism, so that $S$ is isomorphic to a complex torus; McMullen (printed pp. 129 and 136) records that the periods of a one-form on a complex torus are its two generating periods. Forster's §20.8 (printed pp. 165-166) gives the doubly periodic Abel condition $\sum a_{k} \equiv \sum b_{k} (m o d Γ)$ . The example spells out the invariant differential, the two period vectors and the resulting identification $Jac (X) = X$ .

Depends on

Used by

Principal divisor tests via the Abel-Jacobi map Example
The Abel image in its Jacobian Example

Dependency tree · two levels

121 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

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)
Karl Otto Forster, Lectures on Riemann Surfaces, GTM 81, 4th corrected printing (standard reference, not scraped)