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On the number of zeroes of complete elliptic integrals. (Russian) Zbl 0547.14003

Given a family of elliptic curves \(E_ t\) and a 1-form \(\omega\) on \(E_ t\), the author gives an upper bound for the number of zeroes of a period \(I_{\omega}(t)\) of \(\omega\), on certain domains of the parameter, in terms of the degree of the polar divisor of \(\omega\). The proof uses the fact that \(I_{\omega}(t)\) satisfies a Picard-Fuchs equation, and applies to the associated Riccati equation the method developed by A. G. Khovanskij [Sib. Mat. Zh. 25, No.3, 198-203 (1984) and Funkts. Anal. Prilozh. 18, No.2, 40-50 (1984)].
Reviewer: D.Bertrand

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
33E05 Elliptic functions and integrals
32G20 Period matrices, variation of Hodge structure; degenerations
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