Halphen's Transform and Middle Convolution 1
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چکیده
Of special interest are those Lamé equations with finite monodromy group, having therefore algebraic solutions, studied by Baldassarri, Beukers and van der Waall, Chudnovsky and Chudnovsky, Dwork and many others (cf. [2], [4], [8] just to mention some papers). Lamé equations also occur in the context of Grothendieck’s p-curvature conjecture (cf. [8, p. 15]). This conjecture says that if the p-curvature of a differential equation is zero modulo p for almost all primes p then its monodromy group is finite. More generally, it is conjectured that if the p-curvature is globally nilpotent then the differential equation is geometric (also called coming from geometry, Picard-Fuchs) (s. [1, Chap. II §1]). These conjectures are proven by Chudnovsky and Chudnovsky in the Lamé case for n being an integer. In this case the monodromy group is a dihedral group or reducible (s. [8, Thm. 2.1]). Moreover they showed that, for a given exponent scheme, there is only a finite number of Lamé operators that are globally nilpotent (s. [8, Thm. 2.3]). One also knows that if the monodromy group of a Lamé equation is an arithmetic Fuchsian group of signature (1, e) then it comes from geometry. This gives examples of geometric second order differential equation with 4 singularities (s. [8]). In [17] Krammer determined one such example and showed that it is not a (weak) pullback of a hypergeometric differential equation contradicting a conjecture of Dwork that any globally nilpotent second order differential equation on P1/Q̄ has either algebraic solutions, or is a weak pullback of a Gauss hypergeometric differential equation (cf. [17, Section 11]). But there is also the Halphen transform that changes the Lamé equation into another second order differential equation, again a Heun equation. This was used in [8] for n =
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تاریخ انتشار 2009