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Linear forms in the periods of the exponential and elliptic functions. (English) Zbl 0217.04001

Let \(\wp(z)\) denote a Weierstrass \(\wp\)-function, whose invariants \(g_2,g_3\) are algebraic numbers. Assume that \(\wp(z)\) has no complex multiplications. Let \(\omega_1, \omega_2\) denote a pair of fundamental periods for \(\wp(z)\) with \(\mathcal J(\omega_1/ \omega_2) >0\), and let \(\eta_1, \eta_2\) be defined as usual. T. Schneider proved that both \(1, \omega_1, \eta_1\) and \(1, \omega_2, \eta_2\) and \(\omega_1, \omega_2\) are linearly independent over the field of all algebraic numbers. (See for such results: T. Schneider [Einführung in die transzendenten Zahlen. Berlin etc.: Springer (1957; Zbl 0077.04703), pp. 61–62].)
Recently A. Baker proved a result which implies that the numbers \(1, \omega_1, \omega_2\) are linearly independent over the field of algebraic numbers.
In the present paper the author proves the corresponding result for the numbers \(1, \omega_1, \omega_2, 2\pi i\), using the same method and the following consequence of a recent result of J.-P. Serre: Let \(l\) be a prime number and let \(K_l\) be the field generated by the numbers \(\wp(\omega_1/l)\), \(\wp(\omega_2/l)\), \(\wp'(\omega_1/l)\), \(\wp'(\omega_2/l)\) over \(K=\mathbb Q(g_2,g_3)\). Then the degree of \(K_l\) over \(K\) is \(\gg l^4\).
We note that in another paper the author has generalized another result of A. Baker by proving: Any non-vanishing linear form in the numbers \(\omega_1, \omega_2, \eta_1,\eta_2,2\pi i\) is transcendental.
Reviewer: R. Tijdeman

MSC:

11J89 Transcendence theory of elliptic and abelian functions

Citations:

Zbl 0077.04703
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References:

[1] Baker, A.: On the periods of the Weierstrass ?-function. (To appear in the Proceedings of the Rome conference on number theory.) · Zbl 0223.10019
[2] Baker, A.: On the quasi-periods of the Weierstrass ?-function. Nach. Akad. Wiss. Göttingen II. Math. Phys. Klasse, Nr. 16 (1969).
[3] Coates, J.: The transcendence of linear forms in ?1, ?2, ?1, ?2, 2?i. (To appear in Americon Journal of Mathematics). · Zbl 0224.10032
[4] Hardy, G., Wright, E.: An introduction to the theory of numbers. Fourth ed. Oxford: Oxford Univ. Press 1960. · Zbl 0086.25803
[5] Schneider, T.: Einführung in die transzendenten Zahlen. Berlin-Göttingen-Heidelberg: Springer 1957. · Zbl 0077.04703
[6] Serre, J.-P.: Représentations abéliennes modulo1 et applications. (To appear.)
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