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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 $E_2(z) = 1-24 \sum_{n\ge 1} nz^n(1-z^n)^{-1}$, $E_4(z)= 1+240 \sum_{n\ge 1} n^3z^n (1-z^n)^{-1}$ and $E_6(z)= 1-504 \sum_{n\ge 1} n^5z^n (1-z^n)^{-1}$. If $q$ is a complex number with $0< |q |<1$, then the transcendence degree of the field $\bbfQ (q,E_2(q), E_4(q)$, $E_6(q))$ 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 $A\in\bbfC [z,x_1,x_2,x_3]$ is a non-zero polynomial with degrees $\le L$ $(\ge 1)$ in $z$ and $\le M$ $(\ge 1)$ 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 $\bbfC [z,x_1,x_2,x_3]$ 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 {\it D. Bertrand} [Theta functions and transcendence, Madras Number Theory Symposium 1996, The Ramanujan J. Math. (to appear); see also {\it D. Duverney}, {\it Keiji Nishioka, Kuniko Nishioka} and {\it 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 $\sum_{n\ge 0} q^{-n^2}$, where $q$ is an integer $\ge 2$. Other corollaries, for example the transcendence of series like $\sum_{n\ge 1} (1/F_n)^2$, where $(F_n)$ is the Fibonacci sequence, can be found in Transcendence of Jacobi’s theta series and related results (submitted) by {\it D. Duverney}, {\it K. Nishioka}, {\it K. Nishioka} and {\it I. Shiokawa}. We have also to notice that {\it 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 [{\it G. Philibert}, Ann. Inst. Fourier 38, 85-103 (1988; Zbl 0644.10026)]. For an analysis of all these results, see {\it 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)].

11J85Algebraic independence; Gel’fond’s method
11F03Modular and automorphic functions