Geodesic
On the monodromy groups of lifted Euler equations
Glasgow mathematical journal, Tome 25 (1984) no. 1, pp. 47-57

Voir la notice de l'article provenant de la source Cambridge University Press

In [13], Poincaré asked the following question: Which abstract groups can appear as monodromy groups [14] of second order, linear, homogeneous differential equations with meromorphic coefficients (which might depend on one or more parameters) on C? In the present paper, we initiate a classification of monodromy groups of differential equations on compact Riemann surfaces of genus 1. We proceed as follows: Letbe the general Euler equation [1] which has two regular singular points at 0 and λ in the extended complex plane C. Further, let yu(z)(v – 1,2) be an arbitrary but fixed pair of linearly independent solutions to (1) valid in a neighborhood of some ordinary point. Analytic continuation of each solution along a closed loop A in C–{0, λ}, starting and ending at some fixed base point, produces a new solution yv,A (v = 1, 2) which can be expressed aswhere the constants avlx (v1 ν = l,2) in C depend on the homotopy class [A] of A. Clearly, y1,A(z)/y2,A(z) = To(y1(z)/y2(z)) where the Möbius transformation T:w→ (a11W + a12)/(a21W + a22) depends on [A]. The set of all Mobius transformations T belonging to every possible closed loop A in C–{0,λ} forms a group G, called the monodromy group of the Euler equation (1). G is generated by the Mobius transformation belonging to a simple, closed loop Ao encircling 0. Hence, G is cyclic. 
Kuiken, Kathryn; Masterson, John T. On the monodromy groups of lifted Euler equations. Glasgow mathematical journal, Tome 25 (1984) no. 1, pp. 47-57. doi: 10.1017/S0017089500005425
@article{10_1017_S0017089500005425,
     author = {Kuiken, Kathryn and Masterson, John T.},
     title = {On the monodromy groups of lifted {Euler} equations},
     journal = {Glasgow mathematical journal},
     pages = {47--57},
     year = {1984},
     volume = {25},
     number = {1},
     doi = {10.1017/S0017089500005425},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500005425/}
}
TY  - JOUR
AU  - Kuiken, Kathryn
AU  - Masterson, John T.
TI  - On the monodromy groups of lifted Euler equations
JO  - Glasgow mathematical journal
PY  - 1984
SP  - 47
EP  - 57
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500005425/
DO  - 10.1017/S0017089500005425
ID  - 10_1017_S0017089500005425
ER  - 
%0 Journal Article
%A Kuiken, Kathryn
%A Masterson, John T.
%T On the monodromy groups of lifted Euler equations
%J Glasgow mathematical journal
%D 1984
%P 47-57
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500005425/
%R 10.1017/S0017089500005425
%F 10_1017_S0017089500005425

[1] 1.Boyce, W. E. and DiPrima, R. C., Elementary differential equations and boundary value problems (Wiley 1977). Google Scholar

[2] 2.Erdélyi, A. et al. , Higher transcendental functions Bateman Manuscript Project, Vol. 1 (McGraw-Hill, 1953). Google Scholar

[3] 3.Farkas, H. and Kra, I., Riemann surfaces(Springer-Verlag, 1980). Google Scholar | DOI

[4] 4.Gunning, R. C., Special coordinate coverings of Riemann surfaces, Math. Annalen, 170, (1967) 67–86. Google Scholar

[5] 5.Hall, M., The theory of groups (MacMillan 1970). Google Scholar

[6] 6.Hochstadt, H., The functions of mathematical physics (Interscience, 1971). Google Scholar

[7] 7.Kuiken, K., The effect of rational substitutions on the monodromy group of the general nth order equation of the Fuchsian class, Math. Z, 163, (1978) 111–119. Google Scholar | DOI

[8] 8.Lehner, J., A short course in automorphic functions (Holt, Rinehart and Winston, 1966). Google Scholar

[9] 9.Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (Interscience, 1966). Google Scholar

[10] 10.Massey, W. S., Algebraic topology: an introduction (Harcourt, Brace and World, New York, 1967). Google Scholar

[11] 11.Masterson, J. T., Branched affine (or projective) structures on Riemann surfaces, Ph.D. Thesis, (1981) Polytechnic Institute of New York, (to appear). Google Scholar

[12] 12.Nevanlinna, R. and Paatero, V., Introduction to complex analysis (Addison-Wesley, 1962). Google Scholar

[13] 13.Poincaré, H., Sur les groupes des équations linéaires, Acta Math., 4, (1884) 201–312. Google Scholar | DOI

[14] 14.Poole, E. G. C., Theory of linear differential equations(Dover, 1960). Google Scholar

Cité par Sources :