Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 062, 19 pages      arXiv:1011.6410
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

On Algebraically Integrable Differential Operators on an Elliptic Curve

Pavel Etingof a and Eric Rains b
a) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
b) Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

Received April 25, 2011, in final form June 30, 2011; Published online July 07, 2011

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 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).

Key words: finite gap differential operator; monodromy; elliptic Calogero-Moser system.

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