{"paper":{"title":"On the Picard numbers of abelian varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Klaus Hulek, Roberto Laface","submitted_at":"2017-03-17T04:11:33Z","abstract_excerpt":"We study the possible Picard numbers of abelian varieties of given dimension $g$. If $R_g$ denotes the set of realizable Picard numbers, then $R_g$ is bounded by $g^2$. We show that, for $g$ at least $3$, the set $R_g$ always has gaps and we analyze the nature of these gaps. We further prove that the Picard numbers are asymptotically complete in $[1,g^2]$ as $g$ goes to infinity. Finally we show that every Picard number which can be realized over the complex numbers can already be realized by an abelian variety defined over a number field."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05882","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":""},"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"}