One distinct class of homomorphisms of complex tori are called isogenies. These are endomorphisms of complex tori with a non-zero kernel. For example, if we let be an integer, then there is an associated map sending which has kernel isomorphic to . Isomorphic complex tori There is an isomorphism of complex structures on the real vector space and the set and isomorphic tori can be given by a change of basis of their lattices, hence a matrix in . This gives the set of isomorphism classes of complex tori of dimension , , as the Double coset space Note that as a real manifold, this has dimension this is important when considering the dimensions of moduli of Abelian varieties, which shows there are far more complex tori than Abelian varieties. Line bundles and automorphic forms For complex manifolds , in particular complex tori, there is a constructionpg 571 relating the holomorphic line bundles whose pullback are trivial using the group cohomology of . Fortunately for complex tori, every complex line bundle becomes trivial since . Factors of automorphy Starting from the first group cohomology group we recall how its elements can be represented. Since acts on there is an induced action on all of its sheaves, hence on The -action can then be repsented as a holomorphic map . This map satisfies the cocycle condition if for every and . The abelian group of 1-cocycles is called the group of factors of automorphy. Note that such functions are also just called factors. On complex tori For complex tori, these functions are given by functions which follow the cocycle condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map can be written as for which is useful for computing invariants related to the associated line bundle. Line bundles from factors of automorphy Given a factor of automorphy we can define a line bundle on as follows: the trivial line bundle has a -action given by for the factor . Since this action is free and properly discontinuous, the quotient bundle is a complex manifold. Furthermore, the projection induced from the covering projection . This gives a map which induces an isomorphism giving the desired result. For complex tori In the case of complex tori, we have hence there is an isomorphism representing line bundles on complex tori as 1-cocyles