On weights of the $l$-adic representation and arithmetic of Frobenius eigenvalues

S. G. Tankeev

Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 181-218.

Résumé

Let $J$ be an absolutely simple Abelian variety over a number field $k$, $[k:\mathbb Q]\infty$. Assume that $\operatorname{Cent}(\operatorname{End}(J\otimes\overline k))=\mathbb Z$. If the division $\mathbb Q$-algebra $\operatorname{End}^0(J\otimes\overline k)$ splits at a prime number $l$, then the $l$-adic representation is defined by the miniscule weights (microweights) of simple classical Lie algebras of types $A_m$, $B_m$, $C_m$ or $D_m$. If $S$ is a K3 surface over a sufficiently large number field $k\subset\mathbb C$ and the Hodge group $\operatorname{Hg}(S\otimes_k\mathbb C)$ is semisimple, then $S$ has ordinary reduction at each non-Archimedean place of $k$ in some set of Dirichlet density 1. If $J$ is an absolutely simple Abelian threefold of type IV in Albert's classification over a sufficiently large number field, then $J$ has ordinary reduction at each place in some set of Dirichlet density 1.

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