## An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described explicitly: there is a distinguished element such that for any other one has with a line bundle of degree . This is the unique element (up to isomorphism) in with nontrivial global sections. Also , where is a nontrivial extension .

However, since the construction of is done in many steps, there is no explicit description of elements of in the paper. I would like to share a construction, which was explained to me by Vadim Vologodsky.

Let be an indecomposable vector bundle of a rank and degree on an elliptic curve . Let , , . Let be a covering of degree and Galois group . Denote . Then

forLet us prove the statement by induction. First assume that and are coprime. Then since there exists a line sub bundle of of degree (this is because of the equivalence between the category of semi-stable bundles of the slope and the category of torsion sheaves on ). Then one has (since is stable) and we get the result.

Now assume that . Since the degree of is we see that is a semi-stable bundle of degree . As before, there exists a line sub bundle of degree . Thus we have , and thus . Thus we have for a line bundle on with .

The cokernel is indecomposable since is indecomposable, so by induction it is isomorphic to where is a line bundle of degree . Actually, one can see that if then is trivial, and since is indecomposable it can’t be the case. Now it suffices to check that is one dimensional. But it is easy to prove by induction that it is isomorphic to , which is one-dimensional.