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Periods and special values of the hypergeometric series. (English) Zbl 0695.10031

In this article some results about values of the hypergeometric series \[ F(a,b,c;z)=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^ 2}{2}+... \] for a,b,c\(\in {\mathbb{Q}}\) and \(z\in {\bar {\mathbb{Q}}}\) are obtained. Apart from the cases where F(a,b,c;z) is an algebraic function of z, some interesting algebraic values of F(a,b,c;z) have recently been discovered by J. Wolfart. In the present article the extension \({\mathbb{Q}}(z,F(a,b,c;z))/{\mathbb{Q}}(z)\) is identified as a Kummer extension with ramification restriction to a certain set of primes. Some remarks about transcendental values of F(a,b,c;z) and their relationship to values of \(\Gamma\) (z), \(z\in {\mathbb{Q}}\), are also made. The essential tool to obtain these results is the theory of complex multiplication of abelian varieties.
Reviewer: M.Flach

MSC:

11J81 Transcendence (general theory)
14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
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References:

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