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Picard-Fuchs equations, Hauptmoduls and integrable systems. (English) Zbl 0973.34075

Braden, H. W. (ed.) et al., Integrability: The Seiberg-Witten and Whitham equations. Amsterdam: Gordon and Breach Science Publishers. 137-151 (2000).
The paper is a survey of results concerning the use of modular functions in the theory of ordinary differential equations and its applications to mathematical physics. The general solutions to certain differential systems can be expressed in terms of modular functions \(\lambda (\tau)\) where \(\tau\) is the ratio of the elliptic periods. An example is the Darboux-Halphen system \(w_i'=w_i(w_{i+1}+w_{i+2})-w_{i+1}w_{i+2}\), \(i=1,2,3\), \(w_4=w_1\), \(w_5=w_2\). Its general solution is obtained from a known particular one by composing with a general Möbius transformation. When finding the solutions one uses the fact that the period integrals viewed as functions of \(\lambda\) satisfy a certain hypergeometric equation of Legendre type, and \(\lambda (\tau)\) satisfies a Schwarzian equation (i.e. an equation in the Schwarzian derivative). The above hypergeometric equation can be viewed as an example of a Picard-Fuchs one, i.e. a Fuchsian equation determining the variation of elliptic integrals over affine families of elliptic curves. Picard-Fuchs equations arise also as isomonodromic deformation equations, such as Painlevé equation \(P_{VI}\).
Further the paper deals with generalized Halphen equations and with inhomogeneous Picard-Fuchs equations underlying the arising Schwarzian equations. The latter are satisfied by certain Hauptmoduls, i.e. uniformizing functions for Riemann surfaces of genus zero. The inhomogeneous Picard-Fuchs equations and the inhomogeneous Gauss-Manin systems associated to elliptic integrals with varying endpoints are derived. With their help one determines solutions to equations algebraically related to a class of Painlevé VI equations.
For the entire collection see [Zbl 0948.00012].
Reviewer: V.P.Kostov (Nice)

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
81T80 Simulation and numerical modelling (quantum field theory) (MSC2010)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
33E17 Painlevé-type functions
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
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