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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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid."}],"snapshot_sha256":"4b1841f32cc26e386cd6273a265f10568fc2f38b1c47c07cbbd38fd8d71de103"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"90664ec01e446925c24730c8df92c66057fbbe18c5cf19049b2d639c6b564b95"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2603.24340/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Seokhyun Choi","cross_cats":["math.AG"],"headline":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-03-25T14:21:15Z","title":"Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.24340","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-15T00:34:42.963312Z","id":"8fde33a7-aa3b-41cc-bdc6-8b8c1bea596e","model_set":{"reader":"grok-4.3"},"one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid.","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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