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Linear forms in algebraic points of Abelian functions. II

Published online by Cambridge University Press:  24 October 2008

D. W. Masser
Affiliation:
(University of Nottingham)

Extract

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

(1)Baker, A.Linear forms in the logarithms of algebraic numbers. Mathematika 13 (1966), 204–16.CrossRefGoogle Scholar
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(6)Lang, S.Introduction to transcendental numbers (Reading; Addison-Wesley, 1966).Google Scholar
(7)Masser, D. W. Elliptic functions and transcendence. Ph.D. thesis, University of Cambridge, 1974. (Lecture Notes in Mathematics, 437, Springer, 1975.)Google Scholar
(8)Roth, K. F.Rational approximations to algebraic numbers. Mathematika 2 (1955), 120.CrossRefGoogle Scholar
(9)Schneider, T.Einführung in die transzendenten Zahlen (Berlin; Springer-Verlag, 1957).CrossRefGoogle Scholar
(10)Siegel, C. L.Topics in Complex Function Theory, vol. III (New York; Wiley-Interscience, 1973).Google Scholar
(11)Masser, D. W.Linear forms in algebraic points of Abelian functions. I. Proc. Cambridge Philos. Soc. 77 (1975), 499513.CrossRefGoogle Scholar
(12)Bombieri, E. and Lang, S.Analytic subgroups of group varieties. Inventiones math. 11 (1970), 114.CrossRefGoogle Scholar
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