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 $$(\mathrm{NS}(X_{\bar{K}}){\otimes }_{\mathbb{Z}}{\mathbb{Q}}_p{)}^{G_K}⟶H_{\mathrm{et}}^2(X_{\bar{K}},{\mathbb{Q}}_p(1){)}^{G_K}$$ 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 $$\mathrm{Hom}(A,B)\otimes {\mathbb{Q}}_p\to {\mathrm{Hom}}_{G_K}(V_p(A),V_p(B))$$ 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 $(\phi ,N)$-modules. Finally, we disprove all of these conjectures and questions by showing that they can fail for algebraisable abeloid surfaces.

44 pages, final version