{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:G5DF5I2SJJVGYDR2DLYRI27CUF","short_pith_number":"pith:G5DF5I2S","canonical_record":{"source":{"id":"0910.1887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-10T00:50:59Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"fbfcb070cb68b96c209be382b17ee1c0d3eb16e67c3b533f5719d57659ccd8a2","abstract_canon_sha256":"117fd38ed7dc3d9bba7daf4971318bbfa4fedd60ce37f51d1ac0b0e199f2a10f"},"schema_version":"1.0"},"canonical_sha256":"37465ea3524a6a6c0e3a1af1146be2a15e872fed51d8d8953dcdc443af68e31e","source":{"kind":"arxiv","id":"0910.1887","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.1887","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"arxiv_version","alias_value":"0910.1887v2","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1887","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"pith_short_12","alias_value":"G5DF5I2SJJVG","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"G5DF5I2SJJVGYDR2","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"G5DF5I2S","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:G5DF5I2SJJVGYDR2DLYRI27CUF","target":"record","payload":{"canonical_record":{"source":{"id":"0910.1887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-10T00:50:59Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"fbfcb070cb68b96c209be382b17ee1c0d3eb16e67c3b533f5719d57659ccd8a2","abstract_canon_sha256":"117fd38ed7dc3d9bba7daf4971318bbfa4fedd60ce37f51d1ac0b0e199f2a10f"},"schema_version":"1.0"},"canonical_sha256":"37465ea3524a6a6c0e3a1af1146be2a15e872fed51d8d8953dcdc443af68e31e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:22.653217Z","signature_b64":"Zh+HsqV7WMD7IS5vNZIvnQo4XIup+ZoaSLjLDcgKfwBVg3d6gSya4FXAlyQm4RV0DlZR5js7lxe9MEp4ZCSnDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37465ea3524a6a6c0e3a1af1146be2a15e872fed51d8d8953dcdc443af68e31e","last_reissued_at":"2026-05-18T04:31:22.652806Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:22.652806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0910.1887","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E01IlddK8lgSRa4nLj/CRDqrS7vMSQyolNNP1QvY/RhqG4MEaf6r8JWHr+bZkWvWx1e4i7mDtyqCL8CrbWYGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:18:19.659104Z"},"content_sha256":"f4a4862d5a926c4e540df46ab372121bbb5faa40e42c777259e20f57165a4f3a","schema_version":"1.0","event_id":"sha256:f4a4862d5a926c4e540df46ab372121bbb5faa40e42c777259e20f57165a4f3a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:G5DF5I2SJJVGYDR2DLYRI27CUF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exponential Sums and Polynomial Congruences Along p-adic Submanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Dirk Segers, W. A. Zuniga-Galindo","submitted_at":"2009-10-10T00:50:59Z","abstract_excerpt":"In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \\mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1887","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R2c5rKnb+sPdDu6X+2evmFgnsiG9FBY8a42hMC3KalWN8lKQ8K4DNB1PieA8+aZpycd6PXd0nwndM1t37EGAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T12:18:19.659455Z"},"content_sha256":"931495bf6766baf269ef6882c4acc10c897966eaad1003e5b148365a40e4385b","schema_version":"1.0","event_id":"sha256:931495bf6766baf269ef6882c4acc10c897966eaad1003e5b148365a40e4385b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/bundle.json","state_url":"https://pith.science/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T12:18:19Z","links":{"resolver":"https://pith.science/pith/G5DF5I2SJJVGYDR2DLYRI27CUF","bundle":"https://pith.science/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/bundle.json","state":"https://pith.science/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G5DF5I2SJJVGYDR2DLYRI27CUF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:G5DF5I2SJJVGYDR2DLYRI27CUF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"117fd38ed7dc3d9bba7daf4971318bbfa4fedd60ce37f51d1ac0b0e199f2a10f","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-10T00:50:59Z","title_canon_sha256":"fbfcb070cb68b96c209be382b17ee1c0d3eb16e67c3b533f5719d57659ccd8a2"},"schema_version":"1.0","source":{"id":"0910.1887","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.1887","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"arxiv_version","alias_value":"0910.1887v2","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.1887","created_at":"2026-05-18T04:31:22Z"},{"alias_kind":"pith_short_12","alias_value":"G5DF5I2SJJVG","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"G5DF5I2SJJVGYDR2","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"G5DF5I2S","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:931495bf6766baf269ef6882c4acc10c897966eaad1003e5b148365a40e4385b","target":"graph","created_at":"2026-05-18T04:31:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \\mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the","authors_text":"Dirk Segers, W. A. Zuniga-Galindo","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-10T00:50:59Z","title":"Exponential Sums and Polynomial Congruences Along p-adic Submanifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1887","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f4a4862d5a926c4e540df46ab372121bbb5faa40e42c777259e20f57165a4f3a","target":"record","created_at":"2026-05-18T04:31:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"117fd38ed7dc3d9bba7daf4971318bbfa4fedd60ce37f51d1ac0b0e199f2a10f","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-10-10T00:50:59Z","title_canon_sha256":"fbfcb070cb68b96c209be382b17ee1c0d3eb16e67c3b533f5719d57659ccd8a2"},"schema_version":"1.0","source":{"id":"0910.1887","kind":"arxiv","version":2}},"canonical_sha256":"37465ea3524a6a6c0e3a1af1146be2a15e872fed51d8d8953dcdc443af68e31e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37465ea3524a6a6c0e3a1af1146be2a15e872fed51d8d8953dcdc443af68e31e","first_computed_at":"2026-05-18T04:31:22.652806Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:22.652806Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Zh+HsqV7WMD7IS5vNZIvnQo4XIup+ZoaSLjLDcgKfwBVg3d6gSya4FXAlyQm4RV0DlZR5js7lxe9MEp4ZCSnDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:22.653217Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.1887","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4a4862d5a926c4e540df46ab372121bbb5faa40e42c777259e20f57165a4f3a","sha256:931495bf6766baf269ef6882c4acc10c897966eaad1003e5b148365a40e4385b"],"state_sha256":"a7201e0faa13029eeae8d522cd26c096f38e56e25e910038b7e6174b91ed0143"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4fMUR6XU9thFMH0qOxCSmMAhso4EJSoSFCFdJ9ddYDjQ7nwtcfwLtjNkvnQ5EmE1t4R3fdqB2dmUR1Cyehv7Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T12:18:19.661305Z","bundle_sha256":"2e0987f9113a2725a12a13d00cdba092c0a669920919c27b0a4b48d0f9029c43"}}