space (so ), then we can write out an alternating matrix and calculate the associated products on the vectors associated to . These are Then, taking the inner products (with the standard inner product) of these vectors with the vectors we get so if , then We can then directly verify , which holds for the matrix above. For a fixed , we will write the integral form as . Then, there is an associated Hermitian form given by where Semi-character pairs for Hermitian forms For a Hermitian form a semi-character is a map such that hence the map behaves like a character twisted by the Hermitian form. Note that if is the zero element in , so it corresponds to the trivial line bundle , then the associated semi-characters are the group of characters on . It will turn out this corresponds to the group of degree line bundles on , or equivalently, its dual torus, which can be seen by computing the group of characters <blockquote> whose elements can be factored as maps showing a character is of the form for some fixed dual lattice vector . This gives the isomorphism of the set of characters with a real torus. The set of all pairs of semi-characters and their associated Hermitian form , or semi-character pairs, forms a group where This group structure comes from applying the previous commutation law for semi-characters to the new semicharacter : It turns out this group surjects onto and has kernel , giving a short exact sequence This surjection can be constructed through associating to every semi-character pair a line bundle . Semi-character pairs and line bundles For a semi-character pair we can construct a 1-cocycle on as a map defined as The cocycle relation can be easily verified by direct computation. Hence the cocycle determines a line bundle where the -action on is given by Note this action can be used to show the sections of the line bundle are given by the theta functions with factor of automorphy . Sometimes, this is called the canonical factor of automorphy for . Note that because every line bundle has an associated Hermitian form , and a semi-character can be constructed using the factor of automorphy for , we get a surjection Moreover, this is a group homomorphism with a trivial kernel. These facts can all be summarized in the following commutative diagram