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We will consider in this paper the following automorphic integral: $$\\int_{Z_{A}GL_{2}(\\mathbb{F})\\backslash GL_{2}(\\mathbb{A}_{\\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and oth"},"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":"1309.7467","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-28T15:22:56Z","cross_cats_sorted":[],"title_canon_sha256":"6800dd5293d1a993d8d3fe8df7e7532c63b23a540ee81bd633504d8a0c8dfd3b","abstract_canon_sha256":"26d98602e18ef2919b87d88b4eb760890d5295cb228085bc0a90eaa63c40e971"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:22.569933Z","signature_b64":"EoY+uPkbeZnbkvLYZlpOI1fetzcAAEFPbLPSe8ojhyyaP9fa2KFHcB9BInSY5PTBMzhICJnZude5VXwrERR1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7954390ea3c731a5a5011a63b58c974821220a3ca87d3f516625bc50b9364c7","last_reissued_at":"2026-05-18T03:07:22.569148Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:22.569148Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cuspidal part of an Eisenstein series restricted to an index 2 subfield","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yueke Hu","submitted_at":"2013-09-28T15:22:56Z","abstract_excerpt":"Let $\\mathbb{E}$ be a quadratic extension of a number field $\\mathbb{F}$. 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