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On Algebraically Integrable Differential Operators on an Elliptic Curve
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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).
Keywords: finite gap differential operator; monodromy; elliptic Calogero–Moser system. 
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Pavel Etingof; Eric Rains. On Algebraically Integrable Differential Operators on an Elliptic Curve. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a61/

[1] Airault H., McKean H. P., Moser J., “Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem”, Comm. Pure Appl. Math., 30 (1977), 95–148 | DOI | MR | Zbl

[2] Burchnall J. L., Chaundy T. W., “Commutative ordinary differential operators”, Proc. London Math. Soc. Ser. 2, 21 (1923), 420–440 | DOI | Zbl

[3] Braverman A., Etingof P., Gaitsgory D., “Quantum integrable systems and differential Galois theory”, Transform. Groups, 2 (1997), 31–56, arXiv: alg-geom/9607012 | DOI | MR | Zbl

[4] Chalykh O., Etingof P., Oblomkov A., “Generalized Lamé operators”, Comm. Math. Phys., 239 (2003), 115–153, arXiv: math.QA/0212029 | DOI | MR | Zbl

[5] Chalykh O., “Algebro-geometric Schrödinger operators in many dimensions”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 947–971 | DOI | MR | Zbl

[6] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nonlinear equations of Korteweg–de Vries type, finite-band linear operators and Abelian varieties”, Russ. Math. Surv., 31:1 (1976), 59–146 | DOI | MR | Zbl

[7] Drinfel'd V. G., Krichever I. M., Manin Yu. I., Novikov S. P., “Methods of algebraic geometry in contemporary mathematical physics”, Mathematical Physics Reviews, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 1, Harwood Academic, Chur, 1980, 1–54 | MR

[8] Etingof P., Felder G., Ma X., Veselov A., On elliptic Calogero–Moser systems for complex crystallographic reflection groups, arXiv: 1003.4689 | MR

[9] Etingof P., Oblomkov A., Rains E., “Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces”, Adv. Math., 212 (2007), 749–796, arXiv: math.QA/0406480 | DOI | MR | Zbl

[10] Gesztesy F., Weikard R., “Picard potentials and Hill's equation on a torus”, Acta Math., 176 (1996), 73–107 | DOI | MR | Zbl

[11] Gesztesy F., Unterkofler K., Weikard R., “An explicit characterization of Calogero–Moser systems”, Trans. Amer. Math. Soc., 358 (2006), 603–656 | DOI | MR | Zbl

[12] Halphen G. H., Traité des fonctions elliptiques et de leurs applications, v. 2, Paris, 1888

[13] Krichever I. M., “Integration of nonlinear equations by methods of algebraic geometry”, Funct. Anal. Appl., 11:1 (1977), 12–26 | DOI | MR | Zbl

[14] Krichever I. M., “Elliptic solutions of the Kadomcev–Petviashvili equations, and integrable systems of particles”, Funct. Anal. Appl., 14:4 (1980), 282–290 | DOI | MR

[15] Previato E., “Seventy years of spectral curves: 1923–1993”, Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Springer, Berlin, 1996, 419–481 | DOI | MR | Zbl

[16] Segal G., Wilson G., “Loop groups and equations of KdV type”, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl

[17] Treibich A., “Hyperelliptic tangential covers and finite-gap potentials”, Russ. Math. Surv., 56:6 (2001), 1107–1151 | DOI | MR | Zbl

[18] Unterkofler K., “On the solutions of Halphen's equation”, Differential Integral Equations, 14 (2001), 1025–1050 | MR | Zbl

[19] Weikard R., “On commuting differential operators”, Electron. J. Differential Equations, 2000:19 (2000), 11 pp. | MR