Waldschmidt 1997 nature arithmetique valeurs fonctions modulaires

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theoreme_4: Restates Nesterenko's theorem that for every complex or p-adic q with 0 < |q| < 1 the field generated by q and the Ramanujan values P(q), Q(q), R(q) has transcendence degree at least three.

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M. Waldschmidt, Sur la nature arithmétique des valeurs de fonctions modulaires, Séminaire Bourbaki, 49ème année, 1996/97, exposé no. 824 (novembre 1996), Astérisque 245 (1997), 105--140.

The copy read for this card is the numdam copy (record https://www.numdam.org/item/SB_1996-1997__39__105_0/): 37 physical pages, a numdam cover page followed by printed pp. 105--140 (physical p. $n$ is printed p. $103+n$). Its text layer loses most displayed formulas; the introduction and the definition of $\mathcal C$ (pp. 105--106), the statement of Théorème 4 and its consequences (pp. 118--122) and the pages cited from sections 2.3, 2.5 and 3.1 (pp. 123, 126--128) were read on the page images, the rest from the text layer. Provenance: fetched from https://www.numdam.org/item/SB_1996-1997__39__105_0.pdf on 2026-09-17 (UTC), 2,976,159 bytes. Its Numdam cover page prints "© Société mathématique de France, 1997, tous droits réservés." and refers to Numdam's conditions of use (http://www.numdam.org/conditions), every other right reserved.

Contents

The introduction (p. 105) recalls Schneider's 1937 theorem on $j(\tau)$, the 1995 solution of Mahler's question on $J(q)$ by Barré-Sirieix, Diaz, Gramain and Philibert, and states: "En 1996, Nesterenko a démontré que pour tout nombre complexe $q$ satisfaisant $0<|q|<1$, le degré de transcendance sur $\mathbb Q$ du corps $\mathbb Q(q,P(q),Q(q),R(q))$ est au moins égal à 3."

• Section 2.2, "Indépendance algébrique de trois nombres" (p. 118): Théorème 4, "le résultat principal de [2] et [3]", where [2] is Nesterenko's Mat. Sb. paper (card) and [3] his C. R. note. Mahler's theorem that $P,Q,R$ are algebraically independent over $\mathbb C(z)$ and the Ramanujan differential system are recalled.
• Consequences (pp. 119--122): Bertrand's conjecture that $J(q),DJ(q),D^2J(q)$ are algebraically independent for algebraic $q$ with $0<|q|<1$ (p. 119); Corollaire 1, for complex $q$ with $0<|q|<1$ and $J(q)$ algebraic the numbers $q,P(q),\Delta(q)$ are algebraically independent (p. 119); Corollaire 2, for a Weierstrass $\wp$ with algebraic invariants $g_2,g_3$, a nonzero period $\omega$, the corresponding quasi-period $\eta$ and $\tau$ in the upper half-plane the quotient of two fundamental periods, the numbers $e^{2i\pi\tau},\omega/\pi, \eta/\pi$ are algebraically independent; the text after it derives, in the complex-multiplication case, the same for $e^{2i\pi\tau},\omega,\pi$ (p. 119); Corollaire 3, in particular $\pi$ and $e^{\pi}$ are algebraically independent (p. 120); Corollaire 4, on the Jacobi theta series, from which $\theta_3(q)$ is transcendental for algebraic $q$ with $0<|q|<1$, settling the transcendence of $\sum_{n\ge0}\ell^{-n^2}$ for integers $\ell>1$, open since Liouville remarked that his method gave only its irrationality (p. 121); Corollaire 5, an application to Lucas sequences (p. 122).
• Section 2.3, "Mesures d'indépendance algébrique" (p. 123): quantitative refinements of Théorème 4 by Nesterenko and Philippon are reported.
• Section 2.5, "Démonstration du théorème 4" (pp. 126--128): Proposition 3 (p. 127), a sequence of integer polynomials small but nonzero at $(q,P(q),Q(q),R(q))$, combined with Philippon's criterion of algebraic independence, Proposition 4 (pp. 127--128). Nesterenko's zero estimate ([2], Theorem 3) is stated as Théorème 5 in section 3.1 (p. 128).

The exposé does not mention the divisor-sum series $\sum_{n\ge1}\sigma(n)/2^n$ itself; its bearing on Problem 250 is as an independent exposition of the theorem whose Corollary 2 makes that number transcendental.

Compiled scope

Only Théorème 4 is extracted; for complex $q$ it restates Nesterenko's Theorem 1, and the exposé states it for $p$-adic $q$ as well. The exposé's proof sections were not checked. Its role here is exposition: a Bourbaki seminar exposé of November 1996 restating the theorem.

Bears on. #250, as an exposition of the theorem behind the transcendence of the problem's number; it records no new result about the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.