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To do so, we give, using the language of stacks, a modular description of the elliptic homology of $DA(1)$ via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic des"},"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":"1305.6100","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-05-27T03:34:47Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"3be9821d702c0bed198c78df4d9b13ba015ad8fbfb3bf38cc66788fea886770e","abstract_canon_sha256":"87718e49fbc5e91bd88c875457175910130a06e60498aa460ab123eec5afdf56"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:09.085531Z","signature_b64":"9j8f8iSMfrIQWv0UtRRZZHdSEWj6dQekQRlnW5Agi2H4k+EHH6xc8gXsaA7zQG6WP1oJAnHZmcSwpeddJo0gDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0ff5448f9576e9c79d33535b8ca4b191a822c79da41267d77344f582cb63a35","last_reissued_at":"2026-05-18T01:24:09.084791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:09.084791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The homology of $\\mathrm{tmf}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AT","authors_text":"Akhil Mathew","submitted_at":"2013-05-27T03:34:47Z","abstract_excerpt":"We compute the mod $2$ homology of the spectrum $\\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\\mathrm{tmf} \\wedge DA(1) \\simeq \\mathrm{tmf}_1(3) \\simeq BP\\left \\langle 2\\right\\rangle$, where $DA(1)$ is an eight cell complex whose cohomology \"doubles\" the subalgebra $\\mathcal{A}(1)$ of the Steenrod algebra generated by $\\mathrm{Sq}^1$ and $\\mathrm{Sq}^2$. 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