Prime form

For prime forms in music, see Set (music) § Non-serial.
Not to be confused with Schottky form.

In algebraic geometry, the Schottky–Klein prime form E(x,y) of a compact Riemann surface X depends on two elements x and y of X, and vanishes if and only if x = y. The prime form E is not quite a holomorphic function on X × X, but is a section of a holomorphic line bundle over this space. Prime forms were introduced by Friedrich Schottky and Felix Klein.

Prime forms can be used to construct meromorphic functions on X with given poles and zeros. If Σniai is a divisor linearly equivalent to 0, then ΠE(x,ai)ni is a meromorphic function with given poles and zeros.

See also

References

This article is issued from Wikipedia - version of the 10/5/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.