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Assuming some conjectures on the sparsity of newforms of weight $2$ and prime level with unexpectedly high analytic rank, we show that $\\max S(d) \\le 3d + 1$ for sufficiently large even $d$ and $\\max S(d) = o(d)$ for odd $d$."},"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":"2505.14109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-05-20T09:17:27Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"118f9c057b05b6952531a7dcd3ab799a147d76e332809138873c1aca81c104da","abstract_canon_sha256":"6908d66d430cc9735ebf755f37d076a67ee9602dbaf4ffd798b27983d93361d5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:05:56.355769Z","signature_b64":"5PodWEcMlUzvOzTZbFtnhN2nyPmQZ+uJqJa9Ms9JlDAdLIh65yixJvapAdJ+q3ui2p5z3nLiEKXbVILR5lM6DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f9f09b578931d4434e57abffee4d64beedc7b0d68d7095c1db2f0ea988934dc","last_reissued_at":"2026-07-05T11:05:56.355363Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:05:56.355363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Prime order torsion on elliptic curves over number fields. 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