If the ℓ-adic cohomology of a projective smooth variety, defined over a local field K with finite residue field k, is supported in codimension ≥ 1, then every model over the ring of integers of K has a k-rational point. For K a p-adic field, this is proved in (Esnault, 2007, Theorem 1.1). If the model χ is regular, one has a congruence |χ(k)| ≡ 1 modulo |k| for the number of k-rational points (Esnault, 2006, Theorem 1.1). The congruence is violated if one drops the regularity assumption.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Rational point