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On the monotonicity of the quotient of certain abelian integrals. (English) Zbl 0777.34026

The author considers the family of real elliptic curves given by the Legendre normal form, \(y^ 2=x(x-1)(x-s)\), for \(0<s<1\). For each \(s\) the corresponding curve has two connected components, one compact the other not compact. He then investigates the abelian integrals of the differentials \(ydx\) and \(xydx\) over the compact elliptic curve \(\Gamma_ s\) corresponding to \(s\). The method then implemented to prove the quotient of these abelian integrals are bounded and strictly increasing on the interval \(0\leq s\leq 1\) is due to Cashman and Sanders. The method first determines the Picard-Fuchs equation satisfied by the quotient of the abelian integrals. \(\Gamma_ s\) denotes the real elliptic curve in the \(xy\)-plane given by the above equation, \(y^ 2=x(x-s)(x-1)\). It then studies the two components, and the details surrounding the technical development and the strategy implemented are clearly, concisely and well developed. The paper then takes the abelian integrals denoted \({\mathcal A}\) and \({\mathcal B}\) and determines the Picard-Fuchs equation that they satisfy. Again the details are very well presented. The Riccati equation satisfied by the quotient \(\zeta(s)={{\mathcal B}\over {\mathcal A}}\) of the abelian integrals is then determined. The paper concludes with showing that the quotient \(\zeta(s)\) determines a monotonically increasing function of \(s\) on the interval \(0\leq s\leq 1\).

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
14H52 Elliptic curves
14J25 Special surfaces
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