In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Riemann form in the theory of abelian varieties and modular forms, is the following data:

• A lattice Λ in a complex vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
   C^g.
• An alternating bilinear form α from Λ to the integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s 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
  Alternatization
  In mathematics, the notion of alternatization or alternatisation is used to pass from any map to an alternating map.Let S be a set and A an Abelian group...
  
   of the Chern class
  Chern class
  In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
  
   of any factor of automorphy
  Factor of automorphy
  In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G acts on a complex-analytic manifold X. Then, G also acts on the space of holomorphic functions from X to the complex numbers. A function f is termed an automorphic form if...
  
  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.