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Linear forms in algebraic points of Abelian functions. II

  • D. W. Masser (a1)

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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.

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(1)Baker, A.Linear forms in the logarithms of algebraic numbers. Mathematika 13 (1966), 204–16.
(2)Bombieri, E.Algebraic values of meromorphic maps. Inventiones math. 10 (1970), 267–87.
(3)Cassels, J. W. S.An introduction to diophantine approximation. Cambridge Tracts no. 45. (Cambridge University Press 1957.)
(4)Gelfond, A. O.Transcendental and algebraic numbers (New York; Dover, 1960).
(5)Landau, E.Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale (Leipzig; Teubner, 1918).
(6)Lang, S.Introduction to transcendental numbers (Reading; Addison-Wesley, 1966).
(7)Masser, D. W. Elliptic functions and transcendence. Ph.D. thesis, University of Cambridge, 1974. (Lecture Notes in Mathematics, 437, Springer, 1975.)
(8)Roth, K. F.Rational approximations to algebraic numbers. Mathematika 2 (1955), 120.
(9)Schneider, T.Einführung in die transzendenten Zahlen (Berlin; Springer-Verlag, 1957).
(10)Siegel, C. L.Topics in Complex Function Theory, vol. III (New York; Wiley-Interscience, 1973).
(11)Masser, D. W.Linear forms in algebraic points of Abelian functions. I. Proc. Cambridge Philos. Soc. 77 (1975), 499513.
(12)Bombieri, E. and Lang, S.Analytic subgroups of group varieties. Inventiones math. 11 (1970), 114.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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