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 Σn_ia_i is a divisor linearly equivalent to 0, then ΠE(x,a_i)^n_i is a meromorphic function with given poles and zeros.

See also

• Fay's trisecant identity

References

• Fay, John D. (1973), "The prime- form", Theta functions on Riemann surfaces, Lecture Notes in Mathematics, vol. 352, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060090, ISBN 978-3-540-06517-3, MR 0335789
• Baker, Henry Frederick (1995) [1897], Abelian functions, Cambridge Mathematical Library, Cambridge University Press, ISBN 978-0-521-49877-7, MR 1386644
• Mumford, David (1984), Tata lectures on theta. II, Progress in Mathematics, vol. 43, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4578-6, ISBN 978-0-8176-3110-9, MR 0742776