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Congruence for rational points over finite fields and coniveau over local fields

Authors: Hélène Esnault and Chenyang Xu
Journal: Trans. Amer. Math. Soc. 361 (2009), 2679-2688
MSC (2000): Primary 14G15, 14G05
Published electronically: November 18, 2008
MathSciNet review: 2471935
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Abstract: If the $ \ell$-adic cohomology of a projective smooth variety, defined over a local field $ K$ with finite residue field $ k$, is supported in codimension $ \ge 1$, then every model over the ring of integers of $ K$ has a $ k$-rational point. For $ K$ a $ p$-adic field, this is proved in (Esnault, 2007, Theorem 1.1). If the model $ \mathcal{X}$ is regular, one has a congruence $ \vert\mathcal{X}(k)\vert\equiv 1 $ modulo $ \vert k\vert$ for the number of $ k$-rational points (Esnault, 2006, Theorem 1.1). The congruence is violated if one drops the regularity assumption.

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Additional Information

Hélène Esnault
Affiliation: Abteilung von Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany

Chenyang Xu
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Address at time of publication: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Keywords: Rational point, congruence, coniveau
Received by editor(s): June 7, 2007
Received by editor(s) in revised form: August 27, 2007
Published electronically: November 18, 2008
Additional Notes: This work was partially supported by the DFG Leibniz Preis and the American Institute for Mathematics.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.