Riemann form

In

• A lattice Λ in a complex vector space C^g.
• An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:

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
   positive-definite
   .

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

Riemann forms are important because of the following:

• The
  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.

Furthermore, the complex torus C^g/Λ admits the structure of an abelian variety if and only if there exists an alternating bilinear form α such that (Λ,α) is a Riemann form.

References

• Milne, James (1998), Abelian Varieties, retrieved 2008-01-15
• Hindry, Marc; Silverman, Joseph H. (2000), Diophantine Geometry, An Introduction, Graduate Texts in Mathematics, vol. 201, New York,
  MR 1745599{{citation}}: CS1 maint: location missing publisher (link
  )
• MR 0282985
• "Abelian function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Theta-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]