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Rational points over finite fields for regular models of algebraic varieties of Hodge type \(\geq 1\). (English) Zbl 1254.14019

Let \(R\) be a discrete valuation ring of mixed characteristic \((0,p)\), with perfect residue field \(k\) and fraction field \(K\). The main goal of this article is to prove the following:
Theorem. Let \(X\) be a proper and flat \(R\)-scheme, with generic fibre \(X_K\), such that the following conditions hold:
(a) \(X\) is a regular scheme.
(b) \(X_K\) is geometrically connected.
(c) \(H^q(X_K,{\mathcal O}_{X_K})=0\) for all \(q\geq1\).
If \(k\) is finite, then, for any finite extension \(k'\) of \(k\), the number of \(k'\)-rational points of \(X\) satisfies the congruence \[ |X(k')|\equiv 1\quad\mathrm{mod}|k'|. \] As explained in the article’s introduction, theorems of the above type, relating Hodge theoretic properties of \(X_K\) with congruences on the number of points with values in a finite field, have a long history, starting prominently with the Ax-Katz theorem.
Using the expression of the Zeta function of the special fibre \(X_k\) of \(X\) in terms of rigid cohomology and identifying its slope \(<1\) part with Witt vector cohomology, the above theorem can without much effort be reduced to the following deeper
Theorem. Let \(X\) be a regular, proper and flat \(R\)-scheme. Assume that \(H^q(X_K,{\mathcal O}_{X_K})=0\) for some \(q\geq1\). Then \[ H^q(X_k,W{\mathcal O}_{X_k,{\mathbb Q}})=0. \] This theorem is trivial if the condition that \(H^q(X_K,{\mathcal O}_{X_K})=0\) for some \(q\geq1\) is replaced by the condition that \(H^q(X,{\mathcal O}_{X})=0\) for all \(q\geq1\): then one can conclude with the obvious devissage argument.
The strategy in the general case followed here is to use results of \(p\)-adic Hodge theory relating the Hodge and Newton polygons of certain filtered \(F\)-isocrystals on \(k\).
In the case where \(X\) has semistable reduction, this strategy can be carried out straightforwardly: Namely, then the fundamental comparison theorem of Tsuji stating in particular the weak admissibility of the filtered \(F\)-isocrystals on \(k\) assigned to \(X\) by log crystalline cohomology is available.
The true challenge, however, is to go beyond the case of semistable reduction. Using de Jong’s alteration and Tsuji’s extension of the comparison theorems to truncated simplicial schemes, the proof of the above theorems is ultimately reduced to the following:
Theorem. Let \(X, Y\) be two flat, regular \(R\)-schemes of finite type, of the same dimension, and let \(f:X\to Y\) be a projective and surjective \(R\)-morphism, with reduction \(f_k\) over \(\mathrm{Spec} k\). Then, for all \(q\geq0\), the functoriality homomorphism \[ f_k^*:H^q(X_k,W{\mathcal O}_{X_k,{\mathbb Q}})\longrightarrow H^q(Y_k,W{\mathcal O}_{Y_k,{\mathbb Q}}) \] is injective.
This theorem is deduced from the existence of a trace morphism \[ \tau_{i,\pi}:Rf_*W{\mathcal O}_{Y_k,{\mathbb Q}}\longrightarrow W{\mathcal O}_{X_k,{\mathbb Q}} \] (depending on the choice of a factorization \(f=\pi\circ i\) where \(\pi\) is the projection of a projective space over \(X\) on \(X\), and \(i\) is a closed immersion), and the longest part of this article is devoted to the construction of this trace morphism. A first ingredient is a trace morphism \[ \tau_f^*:Rf_*{\mathcal O}_{Y}\longrightarrow {\mathcal O}_{X}. \] Another tool is then the theory of the relative de Rham Witt complex developed by Langer and Zink.
In the final section, a family of examples illustrating the first listed theorem is presented (this family of examples is not covered by the two cases in which the theorem admits short proofs as indicated above).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
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