$p$-Adic Tate Conjectures and Abeloid Varieties

Abstract

We explore Tate-type conjectures over $p$-adic fields, especially a conjecture of W. Raskind [in: Algebra and number theory. Proceedings of the silver jubilee conference, Hyderabad, India, December 11--16, 2003. New Delhi: Hindustan Book Agency. 99--115 (2005; Zbl 1085.14009)] that predicts the surjectivity of $(\text{NS}(X_{\overline{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_{K}}\longrightarrow H_{\text{ét}}^2(X_{\overline{K}},\mathbb{Q}_p(1))^{G_{K}}$ 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 $\text{Hom}(A,B)\otimes\mathbb{Q}_p \,\to\, \text{Hom}_{G_{K}}(V_p(A),V_p(B))$ is surjective if $A$ and $B$ are abeloid varieties over a $p$-adic field.

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, that is, for abelian surfaces with totally degenerate reduction.