*(English)*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 ${E}_{2}\left(z\right)=1-24{\sum}_{n\ge 1}n{z}^{n}{(1-{z}^{n})}^{-1}$, ${E}_{4}\left(z\right)=1+240{\sum}_{n\ge 1}{n}^{3}{z}^{n}{(1-{z}^{n})}^{-1}$ and ${E}_{6}\left(z\right)=1-504{\sum}_{n\ge 1}{n}^{5}{z}^{n}{(1-{z}^{n})}^{-1}$. If $q$ is a complex number with $0<\left|q\right|<1$, then the transcendence degree of the field $\mathbb{Q}(q,{E}_{2}\left(q\right),{E}_{4}\left(q\right)$, ${E}_{6}\left(q\right))$ 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 $A\in \u2102[z,{x}_{1},{x}_{2},{x}_{3}]$ 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 $A(z,{E}_{2}\left(z\right)$, ${E}_{4}\left(z\right)$, ${E}_{6}\left(z\right))$ is at most $2\xb7{10}^{45}L{M}^{3}$. The crucial point is that the Eisenstein series are solutions of a differential system. The principal difficulty is to determine the ideals of $\u2102[z,{x}_{1},{x}_{2},{x}_{3}]$ 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 ${\sum}_{n\ge 0}{q}^{-{n}^{2}}$, 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 $\left({F}_{n}\right)$ 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)].