Monodromy of a Class of Logarithmic Connections on an Elliptic Curve

Francois-Xavier Machu

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

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.

Détail

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

Bibliographie

[1] Anosov D. A., Bolibruch A. A., The Riemann–Hilbert problem, Aspects of Mathematics, E22, Vieweg Verlag, 1994 | MR

[2] Atiyah M. F., “Vector bundles over an elliptic curve”, Proc. London Math. Soc. (3), 7 (1957), 414–452 | DOI | MR | Zbl

[3] Deligne P., Equations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer Verlag, 1970 | MR | Zbl

[4] Diem C., “Families of elliptic curves with genus $2$ covers of degree $2$”, Collect. Math., 57 (2006), 1–25 ; math.AG/0312413 | MR | Zbl

[5] Enolski V. Z., Grava T., “Singular $Z_N$ curves, Riemann–Hilbert problem and modular solutions of the Schlesinger equations”, Int. Math. Res. Not., 2004:32 (2004), 1619–1683 ; math-ph/0306050 | DOI | MR | Zbl

[6] Esnault H., Viehweg E., “Logarithmic de Rham complexes and vanishing theorems”, Invent. Math., 86:1 (1986), 161–194 | DOI | MR | Zbl

[7] Esnault H., Viehweg E., “Semistable bundles on curves and irreducible representations of the fundamental group”, Algebraic Geometry, Hirzebruch 70 (Warsaw, 1998), Contemp. Math., 241, 1999, 129–138 ; math.AG/9808001 | MR | Zbl

[8] Griffiths P. H., Harris J., Principles of Algebraic Geometry, John Wiley Sons, New York, 1978 | MR | Zbl

[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 | MR | Zbl

[11] Mumford D., “Prym varieties. I. Contributions to analysis”, A Collection of papers dedicated to Lipman Bers, Academic Press, New York, 1974, 325–350 | MR

[12] Korotkin D. A., “Isomonodromic deformations in genus zero and one: algebro-geometric solutions and Schlesinger transformations”, Integrable systems: from classical to quantum, CRM Proc. Lecture Notes, 26, Amer. Math. Soc., Providence, RI, 2000, 87–104 | MR | Zbl

[13] Korotkin D. A., “Solution of matrix Riemann–Hilbert problems with quasi-permutation monodromy matrices”, Math. Ann., 329 (2004), 335–364 ; math-ph/0306061 | DOI | MR | Zbl

[14] Lange H., Narasimhan M. S., “Maximal subbundles of rank two vector bundles on curves”, Math. Ann., 266 (1983), 55–72 | DOI | MR | Zbl

[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”, Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, 703–723 | MR

[17] Tu L. W., “Semistable bundles over an elliptic curve”, Adv. Math., 98:11 (1993), 1–26 | MR

[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 | DOI | MR | Zbl

[19] van der Put M., Singer F., Galois theory of linear differential equations, Springer-Verlag, 2003 | MR