{"paper":{"title":"Rational points over finite fields for regular models of algebraic varieties of Hodge type $\\geq 1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.NT"],"primary_cat":"math.AG","authors_text":"H\\'el\\`ene Esnault, Kay R\\\"ulling, Pierre Berthelot","submitted_at":"2010-09-01T13:31:11Z","abstract_excerpt":"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 $\\geq 1$ in positive degrees. Then we show that the number of $k'$-rational points of $X$ satisfies the congruence $|X(k')| \\equiv 1$ mod $|k'|$. Thanks to \\cite{BBE07}, we deduce such congruences from a vanishing theorem for the Witt cohomology groups $H^q(X_k, W\\sO_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0178","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}