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# Linear forms in algebraic points of Abelian functions. I

## Extract

Let Ω be a Riemann matrix whose 2n columns are vectors of Cn. It is well-known (e.g. (10)) that the field of meromorphic functions on Cn with these vectors among their periods is of transcendence degree n over C. More precisely, this field can be written as C(A, B) where A = (A1, …, An) is a vector of algebraically independent functions of the variable z = (z1, …, zn) and B is algebraic over C(A). We shall assume that B is in fact integral of degree d over the ring C[A]. Since the derivatives ∂f/∂zi of a periodic function f(z) are also periodic, the field is mapped into itself by the differential operators ∂/∂zi. Thus there exists a function C(z) in C[A] such that these operators map the ring C[A, B, C−1] into itself.

## References

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(1)Baker, A.Linear forms in the logarithms of algebraic numbers. Mathematika 13 (1966), 204216.
(2)Bombieri, E.Algebraic values of meromorphic maps. Invent. Math. 10 (1970), 267287.
(3)Cassels, J. W. S.An introduction to diophantine approximation. (Cambridge, 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.)
(7)Masser, D. W. Elliptic functions and transcendence. Ph.D. thesis, University of Cambridge, 1974 (to appear in the Springer Lecture Notes series).
(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.)
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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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