Hecke modifications of vector bundles

TL;DR

This paper provides an accessible overview of Hecke modifications of vector bundles, highlighting their significance in mathematics and presenting new explicit classifications for specific cases.

Contribution

It introduces new explicit descriptions of Hecke modifications for rank 2 bundles over the projective line and elliptic curves, not previously documented.

Findings

Classified all Hecke modifications of the trivial rank 2 bundle over a degree 5 point.

Identified all rank 2, degree 0 bundles admitting a specific Hecke modification.

Connected Hecke modifications to their roles in number theory and algebraic geometry.

Abstract

Hecke modifications of vector bundles have played a significant role in several areas of mathematics. They appear in subjects ranging from number theory to complex geometry. This article intends to be a friendly introduction to the subject. We give an overview of how Hecke modifications appear in the literature, explain their origin and their importance in number theory and classical algebraic geometry. Moreover, we report the progress made in describing Hecke modifications explicitly and why these explicit descriptions are important. We describe all the Hecke modifications of the trivial rank $2$ vector bundle over a closed point of degree $5$ in the projective line, as well as all the vector bundles over a certain elliptic curve, which admit a rank $2$ and degree $0$ trace bundle as a Hecke modification. This result is not present in existing literature.