From Wikipedia, the free encyclopedia
In
mathematics , a Riemann form in the theory of
abelian varieties and
modular forms , is the following data:
the real linear extension αR :C g × C g →R of α satisfies αR (iv , iw )=αR (v , w ) for all (v , w ) in C g × C g ;
the associated
hermitian form H (v , w )=αR (iv , w ) + i αR (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 {{
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]