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Higher Chow cycles on Abelian surfaces and a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry $K$-theory in geometry Cycles and subschemes

Published online by Cambridge University Press:  26 March 2014

Ramesh Sreekantan
Ramesh Sreekantan*
Affiliation:
Indian Statistical Institute, Bangalore, India email rameshsreekantan@gmail.com
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Abstract

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We construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

Keywords

higher Chow groupsrational curvesAbelian surfaces

MSC classification

Secondary: 19E15: Algebraic cycles and motivic cohomology 11G25: Varieties over finite and local fields 14C15: (Equivariant) Chow groups and rings; motives 14G20: Local ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 691 - 711
Copyright
© The Author 2014 

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