eo m / 9 40 70 06 v 1 1 3 Ju l 1 99 4 Reduction of the Manin map modulo

For an abelian variety A over a function field K of characteristic zero equipped with a derivation δ : K → K Manin defined in [Man1], [Man2] a remarkable additive map A(K) → V , where V is a vector space over K, which plays an important role in diophantine geometry over function fields. (Cf. [Co] for a “modern” exposition of Manin’s work. Cf. also [B1], [B2] for a different way of introducing this map.) In the “generic case” this map is a “second order non linear differential operator”. In [V1] the second author defined an analogue of this map in the case of elliptic curves over function fields of characteristic p. This analogous map turned out to be of order one. Then the following results were proved in [V1] for ordinary elliptic curves: