## 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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