Riemann form

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 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 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:

• 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.

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, doi:10.1007/978-1-4612-1210-2, ISBN 0-387-98981-1, MR 1745599
• Mumford, David (1970), Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, London: Oxford University Press, MR 0282985
• "Abelian function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Theta-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]