-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 -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 is smooth and projective over a -adic field and has totally degenerate reduction. Sometimes, this is related to -adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether is surjective if and are abeloid varieties over a -adic field.

Using -adic Hodge theory and Fontaine's functors, we reformulate both problems into questions about the interplay of -versus -structures inside filtered -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, -Adic Tate Conjectures and Abeloid Varieties. Doc. Math. 24 (2019), pp. 1879–1934

DOI 10.4171/DM/718