Transcendance et indépendance algébrique: liens entre les points de vue elliptique et modulaire.

Abstract

There exist, now, numerous transcendental and algebraic independence results about elliptic and modular functions i.e. E2, E4, E6 the standard Eisenstein series, j the modular invariant ... (works done by T. Schneider, D. Masser, G.V. Chudnovsky, Y. Nesterenko, P. Philippon ...). Transcendence properties of modular functions have been studied by using their relations with periods of elliptic integrals; and until 1996, all results about these modular functions were corollaries of “elliptic results” (i.e. results established by means of Weierstrass elliptic functions and elliptic curves). With the proof of Mahler-Manin conjecture (1995) and Nesterenko-Philippon works (1996), we can now get new elliptic and exponential results from modular ones (for example this corollary of Nesterenko's paper “π and exp(π) are algebraically independent”, striking result which owes nothing to the exponential function). My aim is twofold: (1) to recall classical links between elliptic and modular functions and to translate algebraic independence results from one setting to the other; (2) to show that this translation suggests a lot of conjectures.