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Algebraic values of \(G\)-functions. (English) Zbl 0753.11024
It is a well-known fact that functions like \(\log z\) and \(\exp z\) assume transcendental values at non-zero algebraic points. This has been proved for a large class of functions satisfying linear differential equations with polynomial coefficients. Surprisingly, J. Wolfart found around 1985 examples of hypergeometric functions assuming algebraic values at a dense set of algebraic arguments. In this paper we study such examples and in particular their \(p\)-adic evaluations. One particular case is \(f(z)=F(1/12,5/12,1/2\mid z)\). It was shown by Wolfart/Beukers that \(f(1323/1331)=(3/4)\root 4 \of {11}\). In this paper we show for the 7- adic evaluation, \(f(1323/1331)_ 7=(1/4)\root 4 \of {11}\). The examples arise as solutions to Picard-Fuchs equations corresponding to families of algebraic curves. Wolfart’s evaluation follows from period relations between members of such a family. However, the non-archimedean evaluations arise from a totally different mechanism of extending isogenies over base-rings.

MSC:
11G07 Elliptic curves over local fields
11J81 Transcendence (general theory)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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