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

2010/09/01 by Pierre Berthelot, Berthelot, Pierre, Hélène Esnault +3 · 1 citation
Computer Science · Mathematics · #11G25 #13F35 #14F30 #14G05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1009.0178

openalex publication_date 2010/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ≥ 1 in positive degrees. Then we show that the number of k'-rational points of X satisfies the congruence |X(k')| ≡ 1 mod |k'|. Thanks to \citeBBE07, we deduce such congruences from a vanishing theorem for the Witt cohomology groups H^q(X_k, W\sO_Xk,\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 → X.

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