Jacobian variety

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The Jacobian variety of an algebraic curve C of genus g ≥ 1 is a particular abelian variety J of dimension g. Analytically, it can be realized as the quotient space V/L, where V is the vector space of all

<math>

 l = \int_{\gamma} (\cdot): \{\mbox{rational differentials on }
   C \mbox{ without poles}\} \longrightarrow \mathbb{C}, 
 \quad \omega \mapsto \int_{\gamma} \omega

</math>

γ a path in C(C), and L is the lattice of all those l with closed path γ.

An important theorem is Abel's theorem.

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