Abstract
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lamé equation. Our main result gives a closed relation between such differential equations and automorphic forms for symplectic groups. Our study is based on techniques concerning with the monodromy of complex differential equations, the Baker–Akhiezer functions and algebraic curves attached to rings of differential operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Lond. Math. Soc. 21, 420–440 (1922)
Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Non-linear equations of Korteweg-de Vries type, finite-zone linear operators and abelian varieties. Uspekhi Math. Nauk 31, 55–136 (1976)
Erdelyi, M., Magnus, W., Oberhettinger, F., Francesco, T.G.: Higher transcendental functions. McGraw-Hill, NY (1955)
Eichler, M., Zagier, D.: The Theory of Jacobi Forms, Birkhäuser, Progress in Math. 55, (1985)
Griffiths, P.A.: Intoroduction to Algebraic Curves, Translations of Math. Monographs 76 Amer. Math. Soc., (1989)
Krichever, I.: Integration of nonlinear equations by the methods of algebraic geometry. Funkt. Anal. Appl. 11, 12–26 (1977)
Krichever, I., Shiota, T.: Soliton equations and the Riemann–Schottky problem. Adv. Lec. Math. 25, 205–258 (2013)
Mulase, M.: Category of vector bundles on algebraic curves and infinite dimensional Grassmanianns. Int. J. Math. 1, 293–342 (1990)
Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg deVries equation and related non-linear equations. In: International Symposium on Algebraic Geometry Kyoto 1977, 115–153 (1977)
McKean, H.P., Moerbeke, P.M.: The spectrum of Hill’s equation. Invent. Math. 30, 217–274 (1975)
Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton (1971)
Shimura, G.: Arithmeticity of the special values of various zeta functions and the periods of Abelian integrals. Sûgaku Expos. 8(1), 17–38 (1995)
Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83(2), 333–382 (1986)
Takemura, K.: Analytic continuation of eigenvalues of Lamé operator. J. Diff. Eq. 228(1), 1–16 (2004)
Wallenberg, G.: Über die Vertauschbarkeit homogener linearer Differentialaus-drücke. Archiv der Math. Phys. 4, 252–268 (1903)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1962)
Ziegler, C.: Jacobi forms of higher degree. Abh. Math. Sem. Univ. Hamburg 59, 191–224 (1989)
Acknowledgements
This work is supported by The JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI” and The Sumitomo Foundation Grant for Basic Science Research Project (No.150108).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ahmed Sebbar.
Rights and permissions
About this article
Cite this article
Nagano, A. On Rings of Differential Operators Derived from Automorphic Forms. Complex Anal. Oper. Theory 12, 377–414 (2018). https://doi.org/10.1007/s11785-017-0663-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-017-0663-7