The Tate conjecture for certain Abelian varieties over finite fields (Q2773285)

In [Compos. Math. 117, 45--76 (1999; Zbl 0985.14010)] the author showed that the Hodge conjecture for all complex Abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields. The present paper extracts from the proof of this result a statement which allows one to deduce the Tate conjecture for the powers of a single Abelian variety over a finite field from the Hodge conjecture of some lifts to characteristic \(0\). To be more precise, let \(A\) be an Abelian variety with many endomorphisms (a generalization of CM-type) over the algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\) and \(A_0\) its reduction modulo some prime of \(\overline{\mathbb{Q}}\) dividing \(p\). Let \(L(A_0)\) denote the Lefschetz group of \(A_0\) and define \(P(A_0)\) to be the smallest algebraic subgroup of \(L(A_0)\) containing a power of a Frobenius endomorphism of \(A_0\).NEWLINENEWLINEThe main result of the paper is the following statement: If \(P(A_0)\) coincides with the intersection of the kernels of all characters \(L(A_0)\to\mathbb{G}_m\), over all algebraic characters of \(L(A)\), then the Tate conjecture holds for all powers of \(A_0\). An immediate consequence is, that if the Hodge conjecture holds for all powers of \(A\) and \(P(A_0)\) coincides with the intersection of \(L(A_0)\) with the Mumford Tate group of \(A\), then the Tate conjecture holds for all powers of \(A_0\). Several examples of Abelian varieties admitting exotic Tate cycles are given for which the Tate conjecture holds.NEWLINENEWLINEAppendix A (which the reviewer found very helpful) summarizes the theory of Abelian varieties of CM-type over \(\mathbb{C}\) and of Abelian varieties over finite fields and how the reduction map relates the two. Appendix B sharpens a result of \textit{L. Clozel} [Ann. Math. (2) 150, 151--163 (1999; Zbl 0995.14018)] relating numerical and homological equivalence for Abelian varieties over finite fields.