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Rational points over finite fields for regular models of algebraic varieties of Hodge type \geq 1

Pierre Berthelot, Hélène Esnault, Kay Rülling

Number 16
Authors Kay Rülling
Hélène Esnault
Year 2010

Let R be a discrete valuation ring of mixed characteristics (0,p), with finite residue field k and fraction field K, let k' be a finite extension of k, and let X be a regular, proper and flat R-scheme, with generic fibre X_K and special fibre X_k. Assume that X_K is geometrically connected and of Hodge type \geq 1 in positive degrees. Then we show that the number of k'-rational points of X satisfies the congruence |X(k')| \equiv 1 mod |k'|. Thanks to \cite{BBE07}, we deduce such congruences from a vanishing theorem for the Witt cohomology groups H^q(X_k, W\sO_{X_k,\Q}), for q > 0. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular R-schemes X and Y of the same dimension, defined by a surjective projective morphism f : Y \to X.

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