Explicitly representing vector bundles over elliptic curves

Algebraic vector bundles are a construction useful for studying the geometry of varieties; they are objects which associate a vector space to each point of the variety in a "polynomial" fashion. These bundles can be explicitly represented via transition matrices, which encode how the vector spaces vary as one moves along the variety. In 1957, Sir Michael Atiyah showed that every indecomposable bundle over a smooth elliptic curve was determined by a point on the curve, and two invariants; the rank and degree. However, his work is not entirely explicit---using his results, we obtain explicit representations of the bundles in terms of transition matrices. As an application, we present a constructive proof of global generation for certain indecomposable bundles over elliptic curves.