{"paper":{"title":"Graph Laplacians, component groups and Drinfeld modular curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mihran Papikian","submitted_at":"2015-05-26T08:58:20Z","abstract_excerpt":"Let $\\frak{p}$ be a prime ideal of $\\mathbb{F}_q[T]$. Let $J_0(\\frak{p})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\\frak{p})$. Let $\\Phi$ be the component group of $J_0(\\frak{p})$ at the place $1/T$. We use graph Laplacians to estimate the order of $\\Phi$ as $\\mathrm{deg}(\\frak{p})$ goes to infinity. This estimate implies that $\\Phi$ is not annihilated by the Eisenstein ideal of the Hecke algebra $\\mathbb{T}(\\frak{p})$ acting on $J_0(\\frak{p})$ once the degree of $\\frak{p}$ is large enough. We also obtain an asymptotic formula for the size of the discriminant of $\\mathbb{T}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06860","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"}