Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T08:49:13.415Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beukers, F. and Wolfart, J.. Algebraic values of hypergeometric functions. In Proceedings of the Durham Conference on Transcendental Numbers 1986 (to appear).Google Scholar
[2]Chevalley, C. and Weil, A.. Über das Verhalten der Integrale 1. Art bei Automorphismen des Funktionskörpers. Abh. Math. Sem. Univ. Hamburg 10 (1934), 358–361.CrossRefGoogle Scholar
[3]Flach, M.. Herleitung der Chowla–Selberg–Formel aus Shimura'schen Periodenrelationen. Arch. Math. (Basel) 47 (1986), 418–421.CrossRefGoogle Scholar
[4]Lang, S.. Elliptic Curves (Addison Wesley, 1973).Google Scholar
[5]Lipman, J.. Desingularisation of two-dimensional schemes. Ann. of Math. 107 (1978), 151–207.CrossRefGoogle Scholar
[6]Raynaud, M.. Specialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38 (1970), 27–76.CrossRefGoogle Scholar
[7]Schappacher, N.. Periods of Hecke Characters. Lecture Notes in Math. vol. 1301 (Springer-Verlag, 1988).CrossRefGoogle Scholar
[8]Schmidt, C. G.. Arithmetik Abelscher Varietäten mit Komplexer Multiplikation. Lecture Notes in Math. vol. 1082 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[9]Serre, J. P. and Tate, J.. Good reduction of abelian varieties. Ann. of Math. 68 (1968), 492–517.CrossRefGoogle Scholar
[10]Shimura, G.. On analytic families of polarized abelian varieties and automorphic functions. Ann. of Math. 78 (1963), 149–192.CrossRefGoogle Scholar
[11]Shimura, G.. Construction of class fields and zeta functions of algebraic curves. Ann. of Math. 85 (1967), 58–159.CrossRefGoogle Scholar
[12]Shimura, G. and Taniyama, Y.. Complex Multiplication of Abelian Varieties (Mathematical Society of Japan, 1961).Google Scholar
[13]Weil, A.. Basic Number Theory, third ed. (Springer-Verlag, 1974).CrossRefGoogle Scholar
[14]Wolfart, J.. Werte hypergeometrischer Funktionen. Invent. Math. 92 (1988), 187–216.CrossRefGoogle Scholar