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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}(\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.06860","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-05-26T08:58:20Z","cross_cats_sorted":[],"title_canon_sha256":"c790ef74999b10a6546dbf64e9916e680e193e7420b9e43138c8a7a837c1e45a","abstract_canon_sha256":"8af4b77170a2d0b1e84c5f0d2b074074ebe6061ebe633be6d36a54bc3a779c3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:09.104554Z","signature_b64":"gYFCbS0FNp5UAfkTA3sEPNjoALBDLNsvoNpXHHsj4iHMcigaCLXS1TXvTfV41wfNBDCWacXp5ormyJxIYbg5Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa13d6e71bf3a748a56fb1ffc3c23d70d4b9458927bb21aaffbe66adc9437703","last_reissued_at":"2026-05-18T00:54:09.104173Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:09.104173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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