Divisor class group

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Definition

Let $X$ be a complex algebraic variety. The divisor class group if $X$ is defined as the quotient of the group of all divisors of $X$ by the group of principal divisors, viz those divisors that occur as the divisor for some meromorphic function (or equivalent notion) on $X$.

Here, a divisor on $X$ is an element in the free Abelian group generated by all codimension-1 subvarieties of $X$. In the particular case that $X$ is a complex algebraic curve, the codimension-1 aubvarieties are simply points, and the divisors of $X$ are simply the integral linear combinations of points in $X$.

Particular cases

Eliptic curve and elliptic curve groups

Further information: Elliptic curve group

For a given elliptic curve over the complex numbers, the divisor class group is the same as what we had defined as the elliptic curve group. To see this, we need to observe that the meromorphic functions on the elliptic curve are the linear functions. The zeroes corresponding to a linear function are the points on the line, and the poles are the point at infinity, counted thrice. Hence, a typical principal divisor look like: $A+B+C-3(O)$ where $O$ is the inflection point at infinity.

Quotienting out by this principal divisor group gives exactly what we describe as the elliptic curve group.

Hyperelliptic curve groups

Further information: Hyperelliptic curve group

Hyperelliptic curve groups, which generalize elliptic curve groups, can also be viewed as divisor class groups: the hyperelliptic curve group is simply the divisor class group for the corresponding hyperelliptic curve.