Prime form

Inalgebraic geometry, the Schottky–Klein prime form E(x,y) of a compact Riemann surface X depends on two elements x and yofX, 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.

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