Jacobian variety


Jacobian variety

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.

References

*
*cite conference | author=J.S. Milne | title=Jacobian Varieties | booktitle=Arithmetic Geometry |publisher=Springer-Verlag|location=New York| year=1986 | pages=pp. 167-212|id=ISBN 0-387-96311-1


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