Thanks, I’m glad people are still finding this blog useful, though I haven’t touched it in years. I’m no longer in mathematics so I don’t have any deep insights on rank 2 bundles on elliptic curves, but the main thing is that it’s a family of planes for each point on the curve. If there’s some vector space they all fit into (there should be for some finite N, though no clue how to bound it) then that means its a map from the elliptic curve to the Grassmannian. I vaguely recall something about vector bundles over abelian group schemes decomposing into direct sums of line bundles, but I can’t verify it. Good luck!

]]>2. I am very much helped.

3. Sir can you please give a hint as how do I think of a rank 2 bundle on an elliptic curve? ]]>

It is the reduction to a pencil that is the nontrivial part of the proof. The rest is just Sard’s theorem. ]]>