Vector Bundles on the Moduli Stack of Elliptic Curves

TL;DR

This paper investigates vector bundles on the moduli stack of elliptic curves over various rings, revealing conditions under which they decompose into line bundles or form complex indecomposable structures.

Contribution

It provides a classification of vector bundles over the moduli stack of elliptic curves for different local rings, including new constructions of indecomposable bundles.

Findings

All vector bundles are sums of line bundles over fields or certain DVRs.

Constructs higher rank indecomposable bundles over 3-local integers.

Shows existence of indecomposable bundles of arbitrary high rank over 2-local integers.

Abstract

We study vector bundles on the moduli stack of elliptic curves over a local ring R. If R is a field or a discrete valuation ring of (residue) characteristic not 2 or 3, all these vector bundles are sums of line bundles. For R the 3-local integers, we construct higher rank indecomposable vector bundles and give a classification of vector bundles that are iterated extensions of line bundles. For R the 2-local integers, we show that there are even indecomposable vector bundles of arbitrary high rank.