Differential operators and rank 2 bundles over elliptic curves

Emma Previato; George Wilson

Compositio Mathematica (1992)

  • Volume: 81, Issue: 1, page 107-119
  • ISSN: 0010-437X

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Previato, Emma, and Wilson, George. "Differential operators and rank $2$ bundles over elliptic curves." Compositio Mathematica 81.1 (1992): 107-119. <http://eudml.org/doc/90129>.

@article{Previato1992,
author = {Previato, Emma, Wilson, George},
journal = {Compositio Mathematica},
keywords = {commuting pair of linear ordinary differential operators; vector bundle; classifications of the smooth elliptic curves; automorphism group; algebra of differential operators; indecomposable and decomposable bundles},
language = {eng},
number = {1},
pages = {107-119},
publisher = {Kluwer Academic Publishers},
title = {Differential operators and rank $2$ bundles over elliptic curves},
url = {http://eudml.org/doc/90129},
volume = {81},
year = {1992},
}

TY - JOUR
AU - Previato, Emma
AU - Wilson, George
TI - Differential operators and rank $2$ bundles over elliptic curves
JO - Compositio Mathematica
PY - 1992
PB - Kluwer Academic Publishers
VL - 81
IS - 1
SP - 107
EP - 119
LA - eng
KW - commuting pair of linear ordinary differential operators; vector bundle; classifications of the smooth elliptic curves; automorphism group; algebra of differential operators; indecomposable and decomposable bundles
UR - http://eudml.org/doc/90129
ER -

References

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  1. [1] M.F. Atiyah: Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452. Zbl0084.17305MR131423
  2. [2] H.F. Baker: Note on the foregoing paper 'Commutative ordinary differential operators', by J.L. Burchnall and T.W. Chaundy, Proc. Royal Soc. London (A) 118 (1928), 584-593. JFM54.0439.02
  3. [3] J.L. Burchnall and T.W. Chaundy:(a) Commutative ordinary differential operators, Proc. London Math. Soc.21 (1923), 420-440;(b) Commutative ordinary differential operators, Proc. Royal Soc. London (A) 118 (1928), 557-583;(c) Commutative ordinary differential operators. II, Proc. Royal Soc. London (A) 134 (1932), 471-485. Zbl0003.25701JFM57.0478.01
  4. [4] P. Dehornoy: Opérateurs différentiels et courbes elliptiques, Comp. Math.43 (1981), 71-99. Zbl0475.14032MR631428
  5. [5] V.G. Drinfeld: On commutative subrings of certain non-commutative rings, Funct. Anal. Appl.11(1) (1977), 11-14 (Russian), 9-12 (English). Zbl0368.14011MR476732
  6. [6] P.G. Grinevich: Rational solutions for the equation of commutation of differential operators, Funct. Anal. Appl.16(1) (1982), 19-24 (Russian), 15-19 (English). Zbl0514.47034MR648805
  7. [7] F.A. Grünbaum: Commuting pairs of linear ordinary differential operators of orders four and six, Physica D31 (1988), 424-433. Zbl0654.47027MR954780
  8. [8] E.L. Ince: Ordinary Differential Equations, Longmans Green and Co., London (1926). JFM53.0399.07
  9. [9] I.M. Krichever: Integration of nonlinear equations by methods of algebraic geometry, Funct. Anal. Appl.11(1) (1977), 15-31 (Russian), 12-26 (English). Zbl0346.35028MR1275725
  10. [10] I.M. Krichever: Commutative rings of ordinary differential operators, Funct. Anal. Appl.12(3) (1978), 20-31 (Russian), 175-185 (English). Zbl0408.34008MR509381
  11. [11] I.M. Krichever and S.P. Novikov: Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili (KP) equation, Funct. Anal. Appl.12(4) (1978), 41-52 (Russian), 276-286 (English). Zbl0404.35082MR515628
  12. [12] I.M. Krichever and S.P. Novikov: Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2, Dokl. Akad. Nauk SSSR247 (1979), 33-37; Soviet Math. Doklady20 (1979), 650-654. Zbl0434.35078MR545939
  13. [13] G. Latham: PhD thesis, Berkeley (1989). 
  14. [14] D. Mumford: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related nonlinear equations, Proc. Int. Symp. on Algebraic Geometry, M. Nagata (ed.), Kinokuniya Book Store, Tokyo (1978). Zbl0423.14007MR578857
  15. [15] E. Previato and G. Wilson: Vector bundles over curves and solutions of the KP equations, Proc. Symp. Pure Math.49 (1989) I, 553-569. Zbl0697.35128MR1013152
  16. [16] G.B. Segal and G. Wilson: Loop groups and equations of KdV type, 'Publ. Math. Inst. Hautes Études Sci.61 (1985), 5-65. Zbl0592.35112MR783348
  17. [17] J.-L. Verdier: Équations différentielles algébriques, Séminaire Bourbaki1977-78, Exposé 512 = Lecture Notes in Math.710, 101-122. Zbl0414.14012MR554217
  18. [18] G. Wilson: Algebraic curves and soliton equations, in Geometry Today, E. Arbarello, C. Procesi and E. Strickland (eds), Birkhaüser, Boston-Basel- Stuttgart (1985), pp. 303-329. Zbl0581.35065MR895160

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