On Algebraically Integrable Differential Operators on an Elliptic Curve

Details On Algebraically Integrable Differential Operators on an Elliptic Curve On Algebraically Integrable Differential Operators on an Elliptic Curve

2010-11-29

arxiv.org/abs/1011.6410v3

SIGMA 7 (2011), 062

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2011.062

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).

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