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Source. Théorème 4, printed p. 118 (physical PDF p. 15), section 2.2 "Indépendance algébrique de trois nombres", read on the page image; it is introduced as "Voici le résultat principal de [2] et [3]", where [2] is Nesterenko, Mat. Sb. 187 (1996), 65--96, and [3] his C. R. note of 1996. The exposé's proof is section 2.5 (pp. 126--128), via Proposition 3 (p. 127) and Philippon's criterion, Proposition 4 (pp. 127--128); the zero estimate behind it is Théorème 5 (section 3.1, p. 128). The proof was not checked here.

Statement

"Soit $q$ un élément de $\mathcal C$ satisfaisant $0<|q|<1$. Alors le degré de transcendance sur $\mathbb Q$ du corps

$$\mathbb Q\big(q,P(q),Q(q),R(q)\big)$$

est supérieur ou égal à 3." Here $\mathcal C$ is either $\mathbb C$ or a field $\mathbb C_p$ with $p$ prime (p. 106), so the statement covers complex and $p$-adic $q$. That is, at least three of $q,P(q),Q(q),R(q)$ are algebraically independent over $\mathbb Q$; for $\mathcal C=\mathbb C$ this is Nesterenko's Theorem 1. The exposé recalls Mahler's theorem that $P,Q,R$ are algebraically independent over $\mathbb C(z)$ and gives the Ramanujan system as $12\,DP/P=P-Q/P$, $3\,DQ/Q=P-R/Q$, $2\,DR/R=P-Q^2/R$ with $D=z\,d/dz$ (pp. 118--119), so that $D$ preserves $\mathbb Q[P,Q,R]$.

Consequences listed by the exposé (pp. 119--122)

Bertrand's conjecture ($J(q),DJ(q),D^2J(q)$ algebraically independent for algebraic $q$ with $0<|q|<1$; more generally, for $q\in\mathcal C$ with $0<|q|<1$, $J(q)\ne0$ and $J(q)\ne1728$, at least three of $q,J(q),DJ(q),D^2J(q)$ algebraically independent); Corollaire 1 ($q,P(q),\Delta(q)$ algebraically independent for complex $q$ with $0<|q|<1$ and $J(q)$ algebraic); Corollaire 2 (elliptic periods and quasi-periods); Corollaire 3 (in particular $\pi$ and $e^{\pi}$ algebraically independent); Corollaire 4 (Jacobi theta series; $\theta_3(q)$ transcendental for algebraic $q$ with $0<|q|<1$); Corollaire 5 (Lucas sequences). The card lists them with page numbers.

Relation to Problem 250

For algebraic $q$ the theorem gives that $P(q),Q(q),R(q)$ are algebraically independent (Nesterenko's Corollary 2); at $q=1/2$, $P(1/2)=1-24\sum_{n\ge1}\sigma(n)/2^n$ is transcendental and so is the problem's number. The exposé does not state this specialization.

Coverage

Statement read on the page image. The role of this page is exposition: a Bourbaki seminar report of November 1996 restating the theorem. No proof is checked here.

Bears on. #250, as a restatement of the theorem behind the transcendence of $\sum_{n\ge1}\sigma(n)/2^n$.