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 Cg.
* An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:

1. the real linear extension αR:Cg × Cg→R 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:

* 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 Cg/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, https://www.jmilne.org/math/CourseNotes/av.html, retrieved 2008-01-15
* Hindry, Marc; Silverman, Joseph H. (2000), Diophantine Geometry, An Introduction, Graduate Texts in Mathematics, 201, New York, doi:10.1007/978-1-4612-1210-2, ISBN 0-387-98981-1
* Mumford, David (1970), Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, London: Oxford University Press
* Hazewinkel, Michiel, ed. (2001), "Abelian function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=A/a010220
* Hazewinkel, Michiel, ed. (2001), "Theta-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=T/t092600

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