Riemann form

In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:

1. the real linear extension α_R:C^g × C^g→R of α satisfies α_R(iv, iw)=α_R(v, w) for all (v, w) in C^g × C^g;
2. the associated hermitian form H(v, w)=α_R(iv, w) + iα_R(v, w) is positive-definite.

(The hermitian form written here is linear in the first variable.)

Riemann forms are important because of the following: