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

• 成果类型: Article
• 署名作者: Berthelot, Pierre; Esnault, Helene; Ruelling, Kay
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.1.8
• 发表日期: 2012
• 页码: 413-508
• 关键词: cohomology; frobenius; THEOREM; complex

摘要

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 vertical bar X(k')vertical bar equivalent to 1 mod vertical bar k'vertical bar. We deduce such congruences from a vanishing theorem for the Witt cohomology groups H-q (X-k, WOx(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 cohoniologies 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.