On Algebraically Integrable Differential Operators on an Elliptic Curve

TL;DR

This paper investigates algebraically integrable differential operators on elliptic curves, classifying specific cases, discovering exotic operators on special curves, and exploring their relation to elliptic Calogero-Moser systems.

Contribution

It provides a classification of higher-order algebraically integrable operators on elliptic curves, including exotic cases and conjectural links to integrable systems.

Findings

Classification of third-order operators with one pole

Discovery of exotic operators on special elliptic curves

Conjectural connection to elliptic Calogero-Moser systems

Abstract

We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special elliptic curves defined over ${\mathbb Q}$ which do not deform to generic elliptic curves. We also study algebraically integrable operators of higher order with several poles and with symmetries, and (conjecturally) relate them to crystallographic elliptic Calogero-Moser systems (which is a generalization of the results of Airault, McKean, and Moser).