Hall algebras and Hecke modifications of vector bundles

TL;DR

This paper studies Hecke modifications of vector bundles on algebraic curves, utilizing Hall algebras to classify these modifications and explore their implications for automorphic forms over finite fields.

Contribution

It introduces a reduction method for classifying Hecke modifications and applies Hall algebra techniques for a complete classification on the projective line over finite fields.

Findings

Reduced classification problem to lower rank bundles

Provided a full classification of Hecke modifications on the projective line

Proved the triviality of unramified toroidal automorphic forms

Abstract

In this article, we investigate Hecke modifications of vector bundles on a smooth projective curve $X$ defined over an arbitrary field. We obtain structural results that allow us to reduce the classification problem of Hecke modifications to the case of vector bundles of lower rank. Moreover, when the base field is a finite field and $X$ is the projective line, we apply the Hall algebra of coherent sheaves to provide a full classification of the Hecke modifications, including their multiplicities. These results are applied to study the space of unramified automorphic forms for $\mathrm{PGL}_n$ over the projective line, leading to a proof that the space of unramified toroidal automorphic forms is trivial.