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Let $A/F$ be a simple abelian variety, $f:A \\rightarrow \\mathbb{P}^n$ be a morphism which is finite onto its image, and $\\Gamma \\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\\mathbb{A}^n \\subseteq \\mathbb{P}^n$ and any finite subset $X \\subseteq f(\\Gamma) \\cap \\mathbb{A}^n$, the energy satisfies $E(X) \\ll \\lvert X \\rvert^2$ and the sumset satisfies $\\lvert X+X \\rvert \\gg \\lvert X \\rvert^2$. We then ask whether"},"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":true},"canonical_record":{"source":{"id":"2603.24340","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-03-25T14:21:15Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"3d8dc27a167bf555e34214c233ff1fc1cab9ddc65b866fbcf5739681c4315a41","abstract_canon_sha256":"c4674e00b8c1092f0d9d2965053149f7a8c123bc9b8d731ad1cb2188aebe5066"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:16.582603Z","signature_b64":"F43xKj2klgAu21pAananbJJbe6ZbcBI2wUkFwPpSk3PLWHlWfB4i6YTWqOZXidLatfp0UqqPsk6DG2YsuUv9BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3d0d3d05696d019033f0426cbe2bc61d4fbefdd23cab8fef6bf33e85ea9bf83","last_reissued_at":"2026-06-02T02:04:16.582050Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:16.582050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid.","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Seokhyun Choi","submitted_at":"2026-03-25T14:21:15Z","abstract_excerpt":"We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \\rightarrow \\mathbb{P}^n$ be a morphism which is finite onto its image, and $\\Gamma \\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\\mathbb{A}^n \\subseteq \\mathbb{P}^n$ and any finite subset $X \\subseteq f(\\Gamma) \\cap \\mathbb{A}^n$, the energy satisfies $E(X) \\ll \\lvert X \\rvert^2$ and the sumset satisfies $\\lvert X+X \\rvert \\gg \\lvert X \\rvert^2$. We then ask whether"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that for any affine chart A^n ⊆ P^n and any finite subset X ⊆ f(Γ) ∩ A^n, the energy satisfies E(X) ≪ |X|^2 and the sumset satisfies |X+X| ≫ |X|^2. We then prove that this holds for arbitrary abelian varieties when f is compatible with the decomposition into simple factors, using the uniform Mordell-Lang conjecture.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The uniform Mordell-Lang conjecture must hold; additionally the morphism f must be finite onto its image and, in the general case, compatible with the simple-factor decomposition of A.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Images of rational points on abelian varieties under morphisms to projective space satisfy E(X) ≪ |X|^2 and |X+X| ≫ |X|^2 for finite subsets X in affine charts.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4b1841f32cc26e386cd6273a265f10568fc2f38b1c47c07cbbd38fd8d71de103"},"source":{"id":"2603.24340","kind":"arxiv","version":3},"verdict":{"id":"8fde33a7-aa3b-41cc-bdc6-8b8c1bea596e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T00:34:42.963312Z","strongest_claim":"We show that for any affine chart A^n ⊆ P^n and any finite subset X ⊆ f(Γ) ∩ A^n, the energy satisfies E(X) ≪ |X|^2 and the sumset satisfies |X+X| ≫ |X|^2. 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