Jacobian variety

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In mathematics, the Jacobian variety of a non-singular algebraic curve C of genus g ≥ 1 is a particular abelian variety J, of dimension g. The curve C is a subvariety of J, and generates J as a group.

Analytically, it can be realized as the quotient space V/L, where V is the vector space of all

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

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

An important theorem regarding Jacobian varieties is Abel's theorem.

[edit] References

  • J.S. Milne (1986). "Jacobian Varieties". Arithmetic Geometry, pp. 167-212, New York: Springer-Verlag. ISBN 0-387-96311-1.


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