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
Let 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.