The p-adic generalized Hodge conjecture for third cohomology

Let $X$ be a variety with totally degenerate reduction, and let $N_{1,L}$ and $N'_{1,L}$ be the enriched monodromy operators defined in Section 4.3. Write $N^1H^3(\overline X,\mathbf Q_p)$ for the first coniveau piece, and let $T^1_1$, $T^{-1}_2$, $T^{3\vee}_0$, and $L^*$ be the groups occurring in the enriched monodromy operators. For each finite extension $L/K$, the notation $[\ker N_{1,L}:T^1_1\otimes_{\mathbf Z}\mathbf Q\to\operatorname{Hom}(T^{3\vee}_0\otimes_{\mathbf Z}\mathbf Q,L^*\otimes_{\mathbf Z}\mathbf Q)]$ denotes the corresponding subspace, and similarly for $N'_{1,L}$. The p-adic generalized Hodge conjecture for $H^3$. One has

$$Gr^{-1}N^1H^3(\overline X,\mathbf Q_p)=$$ $$\sum_{[L:K]<\infty}[\ker N_{1,L}:T^1_1\otimes_{\mathbf Z}\mathbf Q\to\operatorname{Hom}(T^{3\vee}_0\otimes_{\mathbf Z}\mathbf Q,L^*\otimes_{\mathbf Z}\mathbf Q)]\otimes_{\mathbf Q}\mathbf Q_p(-1),$$

and

$$Gr^M_{-1}N^1H^3(\overline X,\mathbf Q_p)=$$ $$\sum_{[L:K]<\infty}[\ker N'_{1,L}:T^{-1}_2\otimes_{\mathbf Z}\mathbf Q\to\operatorname{Hom}(T^{3\vee}_0\otimes_{\mathbf Z}\mathbf Q,L^*\otimes_{\mathbf Z}\mathbf Q)]\otimes_{\mathbf Q}\mathbf Q_p(-2).$$

The conjecture identifies the relevant coniveau pieces with sums of kernels of enriched monodromy operators. The two spaces have the same dimension because the monodromy operator $N:T^{-1}_2\to T^1_1$ is an isogeny; the paper proves the conjecture in some cases for products of three Tate elliptic curves.