pp-Adic Tate Conjectures and Abeloid Varieties

  • Oliver Gregory

    TU München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei München, Germany
  • Christian Liedtke

    TU München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei München, Germany
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Abstract

We explore Tate-type conjectures over pp-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 (NS(XK)ZQp)GKHeˊt2(XK,Qp(1))GK(\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 XX is smooth and projective over a pp-adic field KK and has totally degenerate reduction. Sometimes, this is related to pp-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether Hom(A,B)QpHomGK(Vp(A),Vp(B))\text{Hom}(A,B)\otimes\mathbb{Q}_p \,\to\, \text{Hom}_{G_{K}}(V_p(A),V_p(B)) is surjective if AA and BB are abeloid varieties over a pp-adic field.

Using pp-adic Hodge theory and Fontaine's functors, we reformulate both problems into questions about the interplay of Q\mathbb{Q}-versus Qp\mathbb{Q}_p-structures inside filtered (φ,N)(\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.

Cite this article

Oliver Gregory, Christian Liedtke, pp-Adic Tate Conjectures and Abeloid Varieties. Doc. Math. 24 (2019), pp. 1879–1934

DOI 10.4171/DM/718