{"paper":{"title":"On the distribution of the Picard ranks of the reductions of a $K3$ surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Andreas-Stephan Elsenhans, Edgar Costa, J\\\"org Jahnel","submitted_at":"2016-10-25T10:55:12Z","abstract_excerpt":"We report on our results concerning the distribution of the geometric Picard ranks of $K3$ surfaces under reduction modulo various primes. In the situation that $\\rk \\Pic S_{\\overline{K}}$ is even, we introduce a quadratic character, called the jump character, such that $\\rk \\Pic S_{\\overline\\bbF_{\\!\\frakp}} > \\rk \\Pic S_{\\overline{K}}$ for all good primes, at which the character evaluates to $(-1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07823","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1610.07823/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}