Lame operators with finite monodromy - a combinatorial approach

[摘要] We are describing Lame differential operators with a full set of algebraic solutions. For each finite group G, we are describing the possible values of the degree parameter n such that the Lame operator L-n has the projective monodromy group G. The main technical tool is the combinatorics associated to Belyi functions, ideas that we already used in (Rend. Sem. Mat. Univ. Padova 107 (2002) 191-208) for describing the case n = 1. We also supply proofs to some finiteness properties conjectured by Baldassarri and by Dwork, and we work out an explicit formula for the number of essentially different Lame equations when n = 2. This approach can be generalized for arbitrary degree n (see (Counting Integral Lame Equations by Means of Dessins d'Enfants, arXiv: math. CA/0311510) for n integer). (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-01

[关键词] Lame differential operator;Belyi function;Dessins d'enfant