Deligne’s integrality theorem in unequal characteristic and rational points over finite fields. (English) Zbl 1111.14011

Let \(K\) be a local field with ring of integers \(R\subset K\) and finite residue field \(k\). Lt \(\ell\) be a prime number not dividing \(|k|\). If \(V\) is a variety over \(K\), its \(\ell\)-adic cohomology \(H^m(V\times_K\overline K,\mathbb Q_\ell)\) is said to have coniveau 1 if each class in this group dies in \(H^m(U\times_K\overline K,\mathbb Q_\ell)\) after restriction on a nonempty open subset \(U\subset V\).
The author shows among other things the following: Let \(V\) be an absolutely irreducible, smooth projective variety over \(K\), with a regular projective model \(X\) over \(R\). If its \(\ell\)-adic cohomology \(H^m(V\times_K\overline K,\mathbb Q_\ell)\) has coniveau 1 for all \(m\geq 1\), then the number of rational points of the special fibre \(X \times_Rk\) is \(\equiv 1(\mod|k|)\). For the proof she reduces the problem to showing \(|k|\)-divisibility of the eigenvalues of \(\Phi\), a lifting of the geometric Frobenius of \(k\) in the Deligne-Weil group of \(K\), on \(N^1H^m(V \times_K\overline K,\mathbb Q_\ell)\), the first layer of the coniveau filtration.


14G20 Local ground fields in algebraic geometry
11G25 Varieties over finite and local fields
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