{"paper":{"title":"Spectral Reciprocity for the first moment of triple product $L$-functions and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xinchen Miao","submitted_at":"2024-12-27T18:56:07Z","abstract_excerpt":"Let $F$ be a number field with adele ring $\\mathbb{A}_F$, $\\pi_1, \\pi_2$ be two fixed unitary automorphic representations of $\\mathrm{PGL}_2(\\mathbb{A}_F)$ with finite coprime analytic conductor $\\mathfrak{u}$ and $\\mathfrak{v}$, $\\mathfrak{q},\\mathfrak{l}$ be two coprime integral ideals with $(\\mathfrak{q} \\mathfrak{l}, \\mathfrak{u} \\mathfrak{v})=1$. Following [Zac20], we estimate the first moment of $L(\\frac{1}{2}, \\pi \\otimes \\pi_1 \\otimes \\pi_2)$ twisted by the Hecke eigenvalues $\\lambda_{\\pi}(\\mathfrak{l})$, where $\\pi$ runs over unitary automorphic representations of finite conductor div"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.10418","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/2501.10418/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"}