Annales de la Faculté des Sciences de Toulouse

Summary : A linear independence measure is given for the coordinates of algebraic points of abe-lian functions in two variables. From this an abelian analogue of the Franklin-Schneider theoremis deduced.Let A be a simple abelian variety defined over the field of algebraic numbers and let0 : ~2 -~A be a normalised theta homomorphism (cf. [12], § 1.2). be entire func-tions such that (~~(z),...,~v(z)) forms a system of homogeneous coordinates for the point 0(z) in projective v-space. Put fi : = ~i/~~. Assume that ~~(o) ~ 0 ; then algebraic for all i. A point u in C =~ 0 is by definition an algebraic point of 0 if and only if f.(u) is algebraic forall i. The field of abelian functions associated with 0 is ~ (fi ,...,f~). If (ul,u2) is a non-zero algebraic point of 0, the coordinates ul and u2 are linearlyindependent over the algebraic numbers (cf. [12], Theoreme 3.2.1) ; the proof uses the Schneider- Lang criterion (cf. [5] , Chapter II I,