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Modular functions and transcendence problems. (English. Abridged French version) Zbl 0859.11047
The numbers \(\pi\) and \(e^\pi\) are algebraically independent!
It is a corollary of the main theorem of this very important paper:
Denote by \(E_2\), \(E_4\) and \(E_6\) the Eisenstein series \(E_2(z) = 1-24 \sum_{n\geq 1} nz^n(1-z^n)^{-1}\), \(E_4(z)= 1+240 \sum_{n\geq 1} n^3z^n (1-z^n)^{-1}\) and \(E_6(z)= 1-504 \sum_{n\geq 1} n^5z^n (1-z^n)^{-1}\). If \(q\) is a complex number with \(0< |q |<1\), then the transcendence degree of the field \(\mathbb{Q} (q,E_2(q), E_4(q)\), \(E_6(q))\) is at least 3.
Moreover the author gives a quantitative result (measure of algebraic independence) and says that the same result holds in the \(p\)-adic case. A consequence of the theorem is the following:
Let \(\wp\) be an elliptic function with algebraic invariants \(g_2\) and \(g_3\), \(\omega_1\) and \(\omega_2\) two linearly independent periods, and \(\eta_1\) the quasi-period corresponding to \(\omega_1\). Then the three numbers \(\exp (2i\pi \omega_2/ \omega_1)\), \(\omega_1/ \pi\) and \(\eta_1/ \pi\) are algebraically independent. As a special case the author gets the algebraic independence of, for example, \(\pi\), \(e^\pi\) and \(\Gamma (1/4)\).
Apart from the transcendence construction, the main tool is a zero-estimate:
if \(A\in\mathbb{C} [z,x_1,x_2,x_3]\) is a non-zero polynomial with degrees \(\leq L\) \((\geq 1)\) in \(z\) and \(\leq M\) \((\geq 1)\) in each \(x_i\), then the order at the origin of \(A(z,E_2(z)\), \(E_4(z)\), \(E_6(z))\) is at most \(2.10^{45}LM^3\). The crucial point is that the Eisenstein series are solutions of a differential system. The principal difficulty is to determine the ideals of \(\mathbb{C} [z,x_1,x_2,x_3]\) which are stable under the action of a differential operator related to the differential system. A full proof of the theorem is given by the author in [Modular functions and transcendence Sb. Math. 187, No. 9, 1319-1348 (1996), translation from Mat. Sb. 187, No. 9, 65-96 (1996)].
As pointed out by D. Bertrand [Theta functions and transcendence, Madras Number Theory Symposium 1996, The Ramanujan J. Math. (to appear); see also D. Duverney, Keiji Nishioka, Kuniko Nishioka and I. Shiokawa, Transcendence of Jacobi’s theta series, Proc. Japan Acad. Sci. (to appear)], who gives a very interesting point of view on the theorem and related topics, the theorem can be translated in terms of Jacobi’s theta functions. A nice consequence is the transcendence of \(\sum_{n\geq 0} q^{-n^2}\), where \(q\) is an integer \(\geq 2\). Other corollaries, for example the transcendence of series like \(\sum_{n\geq 1} (1/F_n)^2\), where \((F_n)\) is the Fibonacci sequence, can be found in Transcendence of Jacobi’s theta series and related results (submitted) by D. Duverney, K. Nishioka, K. Nishioka and I. Shiokawa.
We have also to notice that P. Philippon has given a very general theorem (containing Nesterenko’s theorem in both the complex and \(p\)-adic cases) for algebraic independence [Indépendance algébrique et \(K\)-fonctions (to appear)]. In that paper there is also an independent proof of the algebraic independence of \(\pi\), \(e^\pi\) and \(\Gamma (1/4)\): instead of a zero-estimate and a criterion for algebraic independence, he uses a measure of algebraic independence of two numbers [G. Philibert, Ann. Inst. Fourier 38, 85-103 (1988; Zbl 0644.10026)]. For an analysis of all these results, see M. Waldschmidt [Sur la nature arithmétique des valeurs de fonctions modulaires, Sém. Bourbaki 49ème année (1996-1997), Exp. No. 824, Astérisque (to appear) and Transcendance et indépendance algébrique de valeurs de fonctions modulaires, CNTA5 Carleton, Août 1996 (to appear)].

11J85 Algebraic independence; Gel’fond’s method
11F03 Modular and automorphic functions