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 two conditions:

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.

(Note: the hermitian form written here is linear in the first variable, in opposition to the standard definition of this encyclopedia, but in accord with the standard use in this specific subject).

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.

References

• Milne, James (1998), Abelian Varieties, http://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, ISBN 0-387-98981-1, MR1745599 
• Mumford, David (1970), Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, London: Oxford University Press, MR0282985 
• Hazewinkel, Michiel, ed. (2001), "Abelian function", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=A/a010220 
• Hazewinkel, Michiel, ed. (2001), "Theta-function", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=T/t092600