$p$-Adic Tate Conjectures and Abeloid Varieties

Abstract

We explore Tate-type conjectures over $p$-adic fields. We study a conjecture of Raskind that predicts the surjectivity of $$ ({\rm NS}(X_{\bar{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_K} \longrightarrow H^2_{\rm et}(X_{\bar{K}},\mathbb{Q}_p(1))^{G_K} $$ if $X$ is smooth and projective over a $p$-adic field $K$ and has totally degenerate reduction. Sometimes, this is related to $p$-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether $$ {\rm Hom}(A,B)\otimes{\mathbb{Q}}_p \,\to\, {\rm Hom}_{G_K}(V_p(A),V_p(B)) $$ is surjective if $A$ and $B$ are abeloid varieties over $K$. Using $p$-adic Hodge theory and Fontaine's functors, we reformulate both problems into questions about the interplay of $\mathbb{Q}$- versus $\mathbb{Q}_p$-structures inside filtered $(\varphi,N)$-modules. Finally, we disprove all of these conjectures and questions by showing that they can fail for algebraisable abeloid surfaces.