Riemann form

Last updated March 31, 2024

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:

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;
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.

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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 {{citation}}: CS1 maint: location missing publisher (link)
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]

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