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Periods and special values of the hypergeometric series

Published online by Cambridge University Press:  24 October 2008

Matthias Flach
Matthias Flach
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB2 1SB
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Extract

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric series

for rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities like

but it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 3 , November 1989 , pp. 389 - 401
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

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