On the ring of unipotent vector bundles on elliptic curves in positive characteristics

TL;DR

This paper investigates the structure of the ring of unipotent vector bundles on elliptic curves over fields of positive characteristic, revealing it is nonnoetherian and has a complex, infinitely layered spectrum, contrasting with the characteristic zero case.

Contribution

It introduces new results on the ring's structure in positive characteristic using Fourier-Mukai transforms, showing it differs fundamentally from the characteristic zero case.

Findings

The ring is nonnoetherian in positive characteristic.

Contains a subring with infinitely many copies of Spec(Z).

Spectrum exhibits complex infinitesimal gluing at prime p.

Abstract

Using Fourier-Mukai transformations, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a polynomial ring in one variable. It turns out that the situation in characteristic p>0 is completely different and rather bizarre: the ring is nonnoetherian and contains a subring whose spectrum contains infinitely many copies of Spec(Z), which are glued with successively higher and higher infinitesimal identification at the point corresponding to the prime p.