Algebraic varieties over fields with differentiation

Yu. R. Vainberg

Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 389-413

Résumé

It is known that there do not exist algebraic homomorphisms of the multiplicative group of a field $K^*$ into the additive group $K^+$. However, if the field $K$ has a nontrivial differentiation $\alpha$, then the logarithmic derivative gives a homomorphism $K^*\to K^+$, $x\to\frac{\alpha x}x$. Yu. I. Manin observed that for abelian varieties $X$ over a field $K$ with a nontrivial differentiation it is possible to construct homothetic homomorphisms of the group of points $X(K)$ into $K$. The study of such homomorphisms (in particular, the computation of the intersection of their kernels) for varieties over function fields permitted Manin to prove the function field analog of Mordell's conjecture. In this paper we introduce and systematically study a class of functions ($\mathscr D$-functions) encountered in the definition of Manin's map $\mu$. We study the map $\mu$ in the case of varieties over a field of formal power series. Bibliography: 10 titles.