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Cycles in the de Rham cohomology of abelian varieties over number fields

Part of: Arithmetic algebraic geometry (Co)homology theory Abelian varieties and schemes Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  08 March 2018

Yunqing Tang
Yunqing Tang*
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08540, USA email [email protected]
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Abstract

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of $\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

Keywords

absolute Tate cyclesHodge cyclesMumford–Tate conjecturealgebraization theorems

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$ 14F40: de Rham cohomology 14G40: Arithmetic varieties and schemes; Arakelov theory; heights 14K15: Arithmetic ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 4 , April 2018 , pp. 850 - 882
Copyright
© The Author 2018 

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