Riemann form

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

  • A lattice Λ in a complex vector space \mathbb{C}^g
  • A weakly nondegenerate alternating bilinear form α from Λ to the integers satisfying the following condition: Consider the map \beta: (v,w) \mapsto \alpha(iv,w) where the α here is the real linear extension of α from the lattice. Then, β is symmetric positive-definite.

Equivalently, the map (v,w) \mapsto \beta(v,w) + i\alpha(v,w) is a positive-definite Hermitian form.

Riemann forms are important because of the following:

  • The alternatization of the Chern class of any factor of automorphy is a Riemann form.
  • Conversely, given any Riemann form, we can construct a factor of automorphy such that the alternatization of its Chern class is the given Riemann form.
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