Meromorphic Matrix Trivializations of Factors of Automorphy over a Riemann Surface

TL;DR

This paper investigates meromorphic matrix functions and trivializations of factors of automorphy over Riemann surfaces, extending classical results from line bundles to vector bundles, with explicit focus on genus 1 surfaces.

Contribution

It generalizes the Jacobi Inversion Theorem to vector bundles and matrix divisors, providing explicit constructions and improvements for automorphic matrix functions.

Findings

Explicit results for genus 1 Riemann surfaces.

Improved methods for constructing automorphic matrix functions.

Extensions of classical line bundle results to vector bundles.

Abstract

It is a consequence of the Jacobi Inversion Theorem that a line bundle over a Riemann surface M of genus g has a meromorphic section having at most g poles, or equivalently, the divisor class of a divisor D over M contains a divisor having at most g poles (counting multiplicities). We explore various analogues of these ideas for vector bundles and associated matrix divisors over M. The most explicit results are for the genus 1 case. We also review and improve earlier results concerning the construction of automorphic or relatively automorphic meromorphic matrix functions having a prescribed null/pole structure.