Measures of Simultaneous Approximation for Quasi-periods of Abelian Varieties

Theorem 1.1 (Chudnovsky’s theorems). Let Λ = Zω+Zω′ ⊂ C be a lattice with invariants g2, g3, Weierstrass functions ℘, ζ and quasi-periods η, η′, all defined in the usual way (see [Sil94]). Each of the sets below contains at least two algebraically independent numbers : first (1) {g2, g3, η ω , π ω}, (2) {g2, g3, ω, η, ω′, η′}; if we assume g2, g3 ∈ Q̄ : (3) { η ω , π ω}, (4) {ω, ω′, η, η′}; and if, still assuming g2, g3 ∈ Q̄, we consider two complex numbers u and u′, Q-linearly independent and such that ℘(u), ℘(u′) ∈ Q̄ : (5) { η ω , ζ(u)− η ωu}, (6) {u, u′, ζ(u), ζ(u′)}.