Algebraic geometry Algebraic curves Theorem

Abel–Jacobi theorem

Builds on Abel–Jacobi map Jacobian variety Riemann surface Weil divisor Meromorphic function

Statement

Let $X$ be a compact connected Riemann surface of genus $g \ge 1$ , with Jacobian variety $J(X)$ . Fix $P_0 \in X$ , and let $A$ be the Abel–Jacobi map, extended additively to divisors on $X$ .

(Abel) A divisor $D$ of degree $0$ on $X$ is the divisor of a meromorphic function on $X$ if and only if $A(D) = 0$ in $J(X)$ .
(Jacobi inversion) Every point of $J(X)$ equals $A(P_1 + \cdots + P_g) - g A(P_0)$ for some points $P_1, \ldots, P_g \in X$ .

Let $\operatorname{Pic}^0(X)$ be the group of divisors of degree $0$ modulo divisors of meromorphic functions. Then $A$ induces an isomorphism of groups $\operatorname{Pic}^0(X) \to J(X)$ .

Remarks

For the direct half of Abel's theorem, let $D = \operatorname{div}(f)$ and consider the fibres $f^{-1}(t)$ for $t \in \mathbb{P}^1$ , counted with multiplicity. The map $t \mapsto A(f^{-1}(t))$ is a holomorphic map from $\mathbb{P}^1$ to the torus $J(X)$ , so it is constant, and $A(D) = A(f^{-1}(0)) - A(f^{-1}(\infty)) = 0$ . The converse builds $f$ as the exponential of an integral of a meromorphic differential with prescribed residues, whose periods are adjusted using $A(D) = 0$ . Jacobi inversion follows from a dimension count: the map from the $g$ -fold symmetric product of $X$ to $J(X)$ has invertible differential at a general point, by Riemann–Roch. So its image is a compact analytic subset of dimension $g$ in the connected torus $J(X)$ of dimension $g$ , which must be all of $J(X)$ . Example: take $g = 1$ and $X = \C / \Lambda$ with base point $0$ . Abel's theorem says that $\sum n_i [z_i]$ with $\sum n_i = 0$ is the divisor of an elliptic function exactly when $\sum n_i z_i \in \Lambda$ .

Going deeper

For a student

The theorem identifies an algebraic object, the Picard group of degree $0$ , with an analytic one, the torus of periods. This identification is how one proves that the Jacobian is an algebraic variety and how one studies linear systems on curves through the geometry of theta divisors. For an elliptic curve it gives the isomorphism between the curve with its group law and $\C / \Lambda$ . It is the starting point of Brill–Noether theory and of the Torelli theorem.

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Tag	`35KB1QCD3Z`
URI	`m2://35KB1QCD3Z`
Kind	statement, theorem
Writing	W0 stub
MSC	14H40, 30F30
Status	This copy of the site has no build data. Run `m2 status 35KB1QCD3Z`.
About	Abel–Jacobi map, Jacobian variety
Uses	Abel–Jacobi map, Jacobian variety, Riemann surface, Weil divisor, Meromorphic function