On homomorphisms of Abelian schemes.~II

S. G. Tankeev

Izvestiya. Mathematics , Tome 11 (1977) no. 6, pp. 1175-1194

Résumé

Let $k$ be a field of algebraic functions of one variable over the field $\mathbf C$ of complex numbers, let $S$ be the complete smooth model of $k$ over $\mathbf C$, and let $\mathscr I_i\to S$ ($i=1,2$) be the Néron models of Abelian varieties $I_i$ over $k$. Suppose that one of the following conditions holds: 1) The minimal models $\mathscr I_i\to S$ admit compactifications whose degenerate fibers are unions of normally crossing smooth irreducible components, and $$ H^0(S,\mathscr Lie_S(\mathscr I_1)\otimes_{\mathscr O_S}\mathscr Lie_S(\mathscr I_2))=(0). $$ 2) The Abelian variety $I_1$ has totally degenerate reduction at a point $v$ of $k$, i.e. the algebraic group $\mathscr I_{1v}$ is an extension of a finite group by a torus. Then for every prime number $l$ the canonical map $$ \operatorname{Hom}_k(I_1,I_2)\otimes_\mathbf Z\mathbf Z_l\to\operatorname{Hom}_{\operatorname{Gal}(\bar k/k)}(T_l(I_1),T_l(I_2)) $$ is an isomorphism. Bibliography: 17 titles.