Adelic Extension Classes, Atiyah Bundles and Non-Commutative Codes

TL;DR

This paper provides an adelic interpretation of extension classes, explicitly constructs adelic representatives for Atiyah bundles on elliptic curves, and applies these to develop new rank r MDS codes with explicit global sections.

Contribution

It introduces an adelic framework for extension classes, constructs explicit adelic representatives for Atiyah bundles, and applies these to create new MDS codes with explicit local descriptions.

Findings

Adelic interpretation of classical extension classes on curves.

Explicit adelic representatives for Atiyah bundles on elliptic curves.

Construction of rank r MDS codes with explicit global sections.

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].