The paper states
Theorem 1.1
In Milne’s rationality conjecture for abelian varieties, 7 October 2026
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Let A/\Qbar be an abelian variety of dimension d with good reduction A_0/\F at w. Let 0\leq r\leq d, and let
\gamma\in H^{2r}(A(\C),\Q(r))
be a rational Hodge class. For every list of Cartier divisors D_1,\ldots,D_{d-r} on A_0, there is a number q\in\Q such that
\begin{aligned}\Tr_\ell\bigl(\gamma_{0,\ell}
c_{1,\ell}(D_1)\cdots c_{1,\ell}(D_{d-r})\bigr)&=q
&&\text{in }\Q_\ell\quad(\ell\ne p),\\
\Tr_p\bigl(\gamma_{0,p}
c_{1,\mathrm{cris}}(D_1)\cdots
c_{1,\mathrm{cris}}(D_{d-r})\bigr)&=q
&&\text{in }\Cw.\end{aligned}
Here \gamma_{0,\ell} is the smooth proper specialization of the \ell-adic realization, and \gamma_{0,p} is the de Rham–crystalline specialization in
H^{2r}_{\mathrm{cris}}(A_0/W(\F))[1/p](r)
\otimes_{\Kzero}\Cw.
The traces use the top Tate twist and send the class of a zero-cycle to its degree. If r=d, the divisor product is 1.