{"paper":{"title":"The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Thu Ha Trieu","submitted_at":"2024-12-01T17:09:26Z","abstract_excerpt":"We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. We then focus on the relationship between the Mahler measure of four-variable exact polynomials and the special value of the $L$-function of $K3$ surfaces at $s = 4$. This result extends the three-variable case studied in \\cite{Tri23}. Finally, we prove, under Beilinson's conjecture, that the Mahler measure of the polynomial $(x+1)(y+1)(z+1) + t$ is expressed in terms of the Riemann zeta function and the $L$-function of the modular form of weight 3 and level 7."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00893","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/2412.00893/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"}