2002/06/26 by Robert S. Maier · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Nonlinear Waves and Solitons #math-ph #math.CA #math.MP #msc:14H05 #msc:33E10 #msc:34A20
paper · pdf · doi:10.1016/j.jde.2003.06.006
published as J. Differential Equations 198 (2004) 16-34. · 20 pages, elsart document class, no figures
arxiv created 2002/06/26 · openalex publication_date 2004/01/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lamé's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lamé equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.