Proof of the Tate conjecture for products of elliptic curves over finite fields (Q1298174)

Let \(X\) be a smooth projective equidimensional scheme over a finitely generated field \(k.\) Let \({\overline k}\) be a separable closure of \(k\) and put \( {\overline X} = X \times_k {\overline k} .\) Denote by \( CH^m(X) \) the Chow group of codimension \(m\) cycles on \(X.\) \textit{J. Tate} [Arithmetic algebraic Geom., Proc. Conf. Purdue Univ. 1963, 93-110 (1965; Zbl 0213.22804)] has conjectured that the cycle class map is surjective. Let \(k = {\mathbb{F}}_q\) be a finite field with \( q = p^r \) elements and let \( E_1, \dots, E_n \) be elliptic curves over \(k.\) In the note the author gives the proof of the Tate conjecture for products of elliptic curves over \(k:\) For any prime number \(l\) different from \(p\) and any non-negative integer \(m\), the cycle class map \[ CH^m (E_1 \times \ldots \times E_n) \otimes {\mathbb{Q}}_l \rightarrow H^{2m}({\overline E_1} \times \ldots \times {\overline E_n,{\mathbb{Q}}_l(m))^{G_k}} \] is surjective. First he prooves that Tate's conjecture over finite extension of \({\mathbb{F}}_q\) implies the conjecture over \({\mathbb{F}}_q\). Second he uses the theory of Honda-Tate and gives (i) a decomposition of the Frobenius endomorphism of \(E/k\); (ii) three statements which follow from the decomposition. The author finishes by using Chow motives. For the list of other known (and unknown) cases of the Tate conjectures see Sect. 5 (Sect. 1) of the paper of \textit{J. Tate} [Proc. Symp. Pure Math. 55, Pt. 1, 71-83 (1994; Zbl 0814.14009)].