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

  1. the real linear extension αR:Cg × CgR of α satisfies αR(iv, iw)=αR(v, w) for all (v, w) in Cg × Cg;
  2. the associated hermitian form H(v, w)=αR(iv, w) + iαR(v, w) is positive-definite.

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

Riemann forms are important because of the following: