Finite monodromy of Pochhammer equation

Haraoka, Yoshishige

Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 767-810.

Nous considérons le groupe de monodromie $G$ de l’équation différentielle de Pochhammer $𝒫$. Soit ${𝒫}_p$ l’équation réduite modulo un nombre premier $p$. Alors, on montre que $G$ est fini si et seulement si ${𝒫}_p$ admet un système fondamental de solutions polynomiales pour presque tous les nombres premiers.

We consider the monodromy group $G$ of the Pochhammer differential equation $𝒫$. Let ${𝒫}_p$ be the reduce equation modulo a prime $p$. Then we show that $G$ is finite if and only if ${𝒫}_p$ has a full set of polynomial solutions for almost all primes $p$.

[1] F. Beukers, G. Heckman, Monodromy for the hypergeometric function nFn-1, Invent. Math., 95 (1989), 325-354. | MR | Zbl

[2] D.V. Chudnovsky, G.V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations. Lecture Notes in Math. 1135, 52-100, Springer, 1985. | MR | Zbl

[3] Y. Haraoka, Number theoretic study of Pochhammer equation. Publ. Math. de l'Université de Paris VI, Problèmes diophantiens, 93 (1989/1990).

[4] Y. Haraoka, Canonical forms of differential equations free from accessory parameters. to appear in SIAM J. Math. Anal. | Zbl

[5] T. Honda, Algebraic differential equations, INDAM Symposia Math., XXIV (1981), 169-204. | MR | Zbl

[6] E.L. Ince, Ordinary differential equations, New York, 1926.

[7] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé : A modern theory of special functions, Vieweg, 1991. | Zbl

[8] N. Katz, Nilpotent connections and the monodromy theorem : application of a result of Turrittin, Publ. Math. I.H.E.S., 39 (1970), 355-412. | Numdam | MR | Zbl

[9] N. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., 18 (1972), 1-118. | MR | Zbl

[10] E. Landau, Eine Angewendung des Eisensteinschen Satz auf die Theorie der Gausschen Differentialgleichung. J. Reine Angew. Math. 127, 92-102 (1904) ; repreinted in Collected Works, vol. II, 98-108, Thales Verlag, Essen, 1987. | JFM

[11] N. Misaki, Reducibility condition of Pochhammer's equation. Master Thesis, Tokyo Univ., 1973 (in Japanese).

[12] K. Okubo, On the group of Fuchsian equations. Seminar Reports of Tokyo Metropolitan University, 1987.

[13] T. Sasai, On a monodromy group and irreducibility conditions of a fourth order Fuchsian differential system of Okubo type, J. Reine Angew. Math., 299/300 (1978), 38-50. | MR | Zbl

[14] T. Sasai, S. Tsuchiya, On a fourth order Fuchsian differential equation of Okubo type, Funk. Ekvac., 34 (1991), 211-221. | MR | Zbl

[15] K. Takano, E. Bannai, A global study of Jordan-Pochhammer differential equations. Funk. Ekvac., 19 (1976), 85-99. | MR | Zbl

[16] E. Whittaker, G. Watson, Modern Analysis, Cambridge, 1927.

[17] T. Yokoyama, On the structure of connection coefficients for hypergeometric systems, Hiroshima Math. J., 18 (1988), 309-339. | MR | Zbl

[18] T. Yokoyama, On an irreducibility condition for hypergeometric systems, preprint. | Zbl