$p$-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

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

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 $(\phi ,N)$-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.