On Lamé operators which are pull-backs of hypergeometric ones

Author:
Bruno Chiarellotto

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2753-2780

MSC:
Primary 34A20; Secondary 14E20, 14H30, 30F40, 34B30

DOI:
https://doi.org/10.1090/S0002-9947-1995-1308004-4

MathSciNet review:
1308004

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Abstract: We give a method that would allow one to calculate the number of Lamé operators, , , with prescribed finite monodromy and do the calculation for the case . We find a bound for the degree over of the field of definition of the coefficients of a Lamé operator with prescribed finite monodromy and give examples of Lamé operators with finite monodromy. Finally we study Lamé operators with infinite monodromy and generic second order differential operators which are pull-backs of hypergeometric ones under algebraic maps.

**[BA]**F. Baldassarri,*On algebraic solutions of Lamé’s differential equation*, J. Differential Equations**41**(1981), no. 1, 44–58. MR**626620**, https://doi.org/10.1016/0022-0396(81)90052-8**[BA1]**F. Baldassarri,*On second-order linear differential equations with algebraic solutions on algebraic curves*, Amer. J. Math.**102**(1980), no. 3, 517–535. MR**573101**, https://doi.org/10.2307/2374114**[BAT]**-,*Towards a Schwarz list for Lamé differential operators via division points on elliptic curves*, preprint.**[BA-DW]**F. Baldassarri and B. Dwork,*On second order linear differential equations with algebraic solutions*, Amer. J. Math.**101**(1979), no. 1, 42–76. MR**527825**, https://doi.org/10.2307/2373938**[BATE]**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****[BE]**Arnaud Beauville,*Les familles stables de courbes elliptiques sur 𝑃¹ admettant quatre fibres singulières*, C. R. Acad. Sci. Paris Sér. I Math.**294**(1982), no. 19, 657–660 (French, with English summary). MR**664643****[BE-ST]**Jan Stienstra and Frits Beukers,*On the Picard-Fuchs equation and the formal Brauer group of certain elliptic 𝐾3-surfaces*, Math. Ann.**271**(1985), no. 2, 269–304. MR**783555**, https://doi.org/10.1007/BF01455990**[BE-HE]**F. Beukers and G. Heckman,*Monodromy for the hypergeometric function _{𝑛}𝐹_{𝑛-1}*, Invent. Math.**95**(1989), no. 2, 325–354. MR**974906**, https://doi.org/10.1007/BF01393900**[CH-CH]**D. V. Chudnovsky and G. V. Chudnovsky,*Applications of Padé approximations to the Grothendieck conjecture on linear differential equations*, Number theory (New York, 1983–84) Lecture Notes in Math., vol. 1135, Springer, Berlin, 1985, pp. 52–100. MR**803350**, https://doi.org/10.1007/BFb0074601**[CH-CH1]**D. V. Chudnovsky and G. V. Chudnovsky,*Transcendental methods and theta-functions*, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 167–232. MR**1013173****[CH-CH2]**D. V. Chudnovsky and G. V. Chudnovsky,*A random walk in higher arithmetic*, Adv. in Appl. Math.**7**(1986), no. 1, 101–122. MR**834223**, https://doi.org/10.1016/0196-8858(86)90009-6**[CO]**F. Coppi,*Tesi di laurea*, Universitá di Padova, 1992.**[DW]**B. Dwork,*Arithmetic theory of differential equations*, Symposia Mathematica, Vol. XXIV (Sympos., INDAM, Rome, 1979) Academic Press, London-New York, 1981, pp. 225–243. MR**619250****[DW1]**-,*Differential operators with nilpotent**-curvature*, Amer. J. Math. (to appear).**[FO]**Otto Forster,*Lectures on Riemann surfaces*, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York, 1991. Translated from the 1977 German original by Bruce Gilligan; Reprint of the 1981 English translation. MR**1185074****[HO]**Taira Honda,*Algebraic differential equations*, Symposia Mathematica, Vol. XXIV (Sympos., INDAM, Rome, 1979) Academic Press, London-New York, 1981, pp. 169–204. MR**619247****[HU]**Dale Husemoller,*Elliptic curves*, Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 1987. With an appendix by Ruth Lawrence. MR**868861****[KA]**Nicholas M. Katz,*Algebraic solutions of differential equations (𝑝-curvature and the Hodge filtration)*, Invent. Math.**18**(1972), 1–118. MR**0337959**, https://doi.org/10.1007/BF01389714**[KA1]**-,*Nilpotent connections and the monodromy theorem*, Publ. Math. Inst. Hautes Études Sci.**39**(1971), 355-412.**[KA2]**-,*A conjecture in the arithmetic theory groups*, Invent. Math.**87**(1987), 13-61.**[KI]**Tosihusa Kimura,*On Fuchsian differential equations reducible to hypergeometric equations by linear transformations*, Funkcial. Ekvac.**13**(1970/71), 213–232. MR**0301271****[MA]**William S. Massey,*A basic course in algebraic topology*, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991. MR**1095046****[PL]**E. Poole,*Introduction to the theory of linear differential equations*, Oxford, 1936.**[SE]**Leon Greenberg,*Maximal Fuchsian groups*, Bull. Amer. Math. Soc.**69**(1963), 569–573. MR**0148620**, https://doi.org/10.1090/S0002-9904-1963-11001-0**[SE-TR]**Herbert Seifert and William Threlfall,*Seifert and Threlfall: a textbook of topology*, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With “Topology of 3-dimensional fibered spaces” by Seifert; Translated from the German by Wolfgang Heil. MR**575168****[SI]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****[SIN]**Michael F. Singer,*Algebraic solutions of 𝑛th order linear differential equations*, Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979) Queen’s Papers in Pure and Appl. Math., vol. 54, Queen’s Univ., Kingston, Ont., 1980, pp. 379–420. MR**634699****[TR]**Marvin Tretkoff,*Algebraic extensions of the field of rational functions*, Comm. Pure Appl. Math.**24**(1971), 491–497. MR**0280467**, https://doi.org/10.1002/cpa.3160240404**[WW]**E. T. Whittaker and G. N. Watson,*A course in modern analysis*, Cambridge Univ. Press, 1927.

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1308004-4

Article copyright:
© Copyright 1995
American Mathematical Society