On the arithmetic nature of the values of modular functions. (Sur la nature arithmétique des valeurs de fonctions modulaires.) (French) Zbl 0908.11029

Waldschmidt, Michel

Séminaire Bourbaki. Volume 1996/97. Exposés 820–834. Paris: Société Mathématique de France, Astérisque. 245, 105-140, Exp. No. 824 (1997).

This is a very useful survey – including some new proofs – about old and new transcendence and algebraic independence results concerning automorphic functions. The author starts with Th. Schneider’s [Math. Ann. 113, 1–13 (1936; Zbl 0014.20402)] transcendence result concerning \(j(\tau)\) where \(j\) is the elliptic modular function and \(\tau\) algebraic but not imaginary quadratic. Then he describes a proof of the Mahler-Manin conjecture that \[ J(q):= J(e^{2\pi i\tau}):= j(\tau) \] has transcendental values at algebraic arguments \(q= e^{2\pi i\tau}\), \(0<| q|< 1\), proved recently by K. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert [Invent. Math. 124, 1–9 (1996; Zbl 0853.11059)].
The second part treats results about algebraic independence of periods of elliptic curves and the values of modular forms and functions at the corresponding arguments in the upper half plane, beginning with the results of G. V. Chudnovsky [Proc. Int. Cong. Math., Helsinki 1978, Vol. 1, 339–350 (1980; Zbl 0431.10019)]. It culminates with Yu. Nesterenko’s result [Sb. Math. 187, No. 9, 1319–1348 (1996); translation from Mat. Sb. 187, 65–96 (1996; Zbl 0898.11031)] that the transcendence degree of \(\mathbb{Q}(q,P(q),Q(q), R(q))\) is at least \(3\) if \(0<| q|<1\) and \(P\), \(Q\), \(R\) denote the Eisenstein series of weight \(2\), \(4\), \(6\) in Ramanujan’s notation. Nestenko’s theorem has many spectacular consequences, e.g. the algebraic independence of the three numbers \(\pi\), \(e^\pi\), \(\Gamma(1/4)\) or \(\pi\), \(e^{\pi\sqrt 3}\), \(\Gamma(1/3)\), several corollaries about values of theta functions, and the transcendence of the series \(\sum F^{-2}_n\) for the Fibonacci numbers \(F_n\). All these results have effective versions!
Besides classical methods of transcendence, the main new ingredients are zero estimates established in the last years by Nesterenko, Philippon and Philibert. These are described in Section 3.
Waldschmidt’s survey ends with a rich commented bibliography (of 83 titles). Many conjectures spread all over the text show that transcendence has made remarkable progress but did not at all arrive at an end.
For the entire collection see [Zbl 0910.00034].

Reviewer: Jürgen Wolfart (Frankfurt am Main)

MSC:

11J89	Transcendence theory of elliptic and abelian functions
11J85	Algebraic independence; Gel’fond’s method
11J91	Transcendence theory of other special functions
11F11	Holomorphic modular forms of integral weight
33B15	Gamma, beta and polygamma functions
11F03	Modular and automorphic functions

Keywords:

survey; transcendence; algebraic independence; automorphic functions; elliptic modular function; Mahler-Manin conjecture; Eisenstein series