Milne's rationality conjecture for abelian varieties

Claims Theorem 1.1: for every abelian variety over $\overline{\mathbb Q}$ with good reduction at a $p$-adic place, every residue characteristic $p$ including $2$, every rational Hodge class $\gamma$ of codimension $r$ and every divisors $D_1,\dots,D_{d-r}$ on the reduction, the $\ell$-adic traces ($\ell\ne p$) and the crystalline trace of $\gamma_0\cdot D_1\cdots D_{d-r}$ equal one rational number. Also claims each specialized Hodge class lies in one fixed good theory of rational Tate classes, and (using the CM Hodge theorem) is represented by a rational algebraic cycle in all realizations. It does not prove the Hodge or Tate conjecture for general abelian varieties.

Claims Theorem 1.1: for every abelian variety over $\overline{\mathbb Q}$ with good reduction at a $p$-adic place, every residue characteristic $p$ including $2$, every rational Hodge class $\gamma$ of codimension $r$ and every divisors $D_1,\dots,D_{d-r}$ on the reduction, the $\ell$-adic traces ($\ell\ne p$) and the crystalline trace of $\gamma_0\cdot D_1\cdots D_{d-r}$ equal one rational number. Also claims each specialized Hodge class lies in one fixed good theory of rational Tate classes, and (using the CM Hodge theorem) is represented by a rational algebraic cycle in all realizations. It does not prove the Hodge or Tate conjecture for general abelian varieties.