{"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. 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.","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","weakest_assumption":"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.","pith_extraction_headline":"Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.24340/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"90664ec01e446925c24730c8df92c66057fbbe18c5cf19049b2d639c6b564b95"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}