{"paper":{"title":"To be or not to be local","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Benjamin Schraen, Christophe Breuil, Florian Herzig, Karol Koziol, Stefano Morra, Sug Woo Shin, Yongquan Hu","submitted_at":"2026-06-27T09:03:37Z","abstract_excerpt":"Let $p$ be a prime number and $K$ a finite unramified extension of $\\mathbf{Q}_p$. For a smooth representation $\\pi$ of $\\mathrm{GL}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K=\\mathbf{Q}_p$) to attack the locality question: does $\\pi$ depend only on the underlying $2$-dimensional representation $\\overline{\\rho}$ of ${\\rm Gal}(\\overline K/K)$? In particular when $[K:\\mathbf{Q}_p]=2$, crucially using perfectoid geometry, we associate to $\\overline{\\rho}$ an infinite-dimensional mod $p$ smooth rep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28818","kind":"arxiv","version":1},"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/2606.28818/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"}