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

SIGMA 3 (2007), 082, 31 pages      arXiv:0708.2186

Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

Francois-Xavier Machu
Mathématiques - bât. M2, Université Lille 1, F-59655 Villeneuve d'Ascq Cedex, France

Received March 22, 2007, in final form August 06, 2007; Published online August 16, 2007

The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their monodromy, differential Galois group and the underlying rank-2 vector bundle. The latter is described in terms of elementary transforms. The question of its (semi)-stability is addressed.

Key words: elliptic curve; ramified covering; logarithmic connection; bielliptic curve; genus-2 curve; monodromy; Riemann-Hilbert problem; differential Galois group; elementary transformation; stable bundle; vector bundle.

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  1. Anosov D.A., Bolibruch A.A., The Riemann-Hilbert problem, Aspects of Mathematics, Vol. 22, Vieweg Verlag, 1994.
  2. Atiyah M.F., Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414-452.
  3. Deligne P., Equations différentielles à points singuliers réguliers, Lecture Notes in Math., Vol. 163, Springer Verlag, 1970.
  4. Diem C., Families of elliptic curves with genus 2 covers of degree 2, Collect. Math. 57 (2006), 1-25, math.AG/0312413.
  5. Enolski V.Z., Grava T., Singular ZN curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equations, Int. Math. Res. Not. 2004 (2004), no. 32, 1619-1683, math-ph/0306050.
  6. Esnault H., Viehweg E., Logarithmic de Rham complexes and vanishing theorems, Invent. Math. 86 (1986), no. 1, 161-194.
  7. Esnault H., Viehweg E., Semistable bundles on curves and irreducible representations of the fundamental group, in Algebraic Geometry, Hirzebruch 70 (Warsaw, 1998), Contemp. Math. 241 (1999), 129-138, math.AG/9808001.
  8. Griffiths P.H., Harris J., Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978.
  9. Jacobi C., Review of Legendre, Théorie des fonctions elliptiques, Troisième supplément, J. Reine Angew. Math. 8 (1832), 413-417.
  10. Katz M.N., A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), 203-239.
  11. Mumford D., Prym varieties. I. Contributions to analysis, in A Collection of papers dedicated to Lipman Bers, Academic Press, New York, 1974, 325-350.
  12. Korotkin D.A., Isomonodromic deformations in genus zero and one: algebrogeometric solutions and Schlesinger transformations. Integrable systems: from classical to quantum, CRM Proc. Lecture Notes, Vol. 26, Amer. Math. Soc, Providence, RI, 2000, 87-104.
  13. Korotkin D.A., Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann. 329 (2004), 335-364, math-ph/0306061.
  14. Lange H., Narasimhan M.S., Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), 55-72.
  15. Loray F., van der Put M., Ulmer F., The Lamé family of connections on the projective line, ccsd-00005796.
  16. Shaska T., Völklein H., Elliptic subfields and automorphisms of genus 2 function fields, in Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, 703-723.
  17. Tu L.W., Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), no. 11, 1-26.
  18. van der Put M., Galois theory of differential equations, algebraic groups and Lie algebras, in Differential Algebra and Differential Equations, J. Symbolic Comput. 28 (1999), 441-472.
  19. van der Put M., Singer F., Galois theory of linear differential equations, Springer-Verlag, 2003.

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