Algebraic independence and transcendence: Links between elliptic and modular viewpoints

There exist, now, numerous transcendental and algebraic independence result s about elliptic and modular functions i.e. E-2, E-4, E-6 the standard Eise nstein series, j the modular invariant ... (works done by T. Schneider, D. Masser, G.V. Chudnovsky, Y. Nesterenko, P. Philippon ...). Transcendence pr operties of modular functions have been studied by using their relations wi th periods of elliptic integrals; and until 1996, all results about these m odular functions were corollaries of "elliptic results" (i.e. results estab lished by means of Weierstrass elliptic functions and elliptic curves). Wit h the proof of Mahler-Manin conjecture (1995) and Nesterenko-Philippon work s (1996), we can now get new elliptic and exponential results from modular ones (for example this corollary of Nesterenko's paper "pi and exp(pi) are algebraically independent", striking result which owes nothing to the expon ential function). My aim is twofold: (1) to recall classical links between elliptic and modular functions and to translate algebraic independence resu lts from one setting to the other; (2) to show that this translation sugges ts a lot of conjectures.