Adelic Extension Classes, Atiyah Bundles and Non-Commutative Codes

Abstract: This paper consists of three components. In the first, we give an adelic interpretation of the classical extension class associated to extension of locally free sheaves on curves. Then, in the second, we use this construction on adelic extension classes to write down explicitly adelic representors in $GL_r(A)$ for Atiyah bundles $I_r$ on elliptic curves. All these works make sense over any base fields. Finally, as an application, for $m\geq 1$, we construct the global sections of $I_r(mQ)$ in local terms and apply it to obtain rank $r$ MDS codes based on the codes spaces $C_{F;r}(D; I_r(mQ))$ introduced in our earlier paper [Codes and Stability].