Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General Case

Seokhyun Choi

arxiv: 2606.01299 · v1 · pith:SWUNDHRYnew · submitted 2026-05-31 · 🧮 math.NT · math.AG

Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General Case

Seokhyun Choi This is my paper

Pith reviewed 2026-06-28 16:18 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords abelian varietiesMordell-Lang conjectureadditive energysumsetsrational pointsarithmetic geometryUeno locus

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The pith

Images of finite-rank subgroups of abelian varieties satisfy E(X) ≪ |X|^2 in affine charts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that, assuming the uniform Mordell-Lang conjecture, the image of a finite-rank subgroup of an abelian variety under a morphism finite onto its image cannot exhibit strong additive structure inside any affine chart of projective space. For any finite subset X drawn from such an image in A^n, the additive energy E(X) is bounded above by a constant times |X| squared while the sumset |X+X| is bounded below by a constant times |X| squared. This shows that the projected points behave like unstructured sets rather than arithmetic progressions or other low-energy configurations. The argument removes the simplicity hypothesis from an earlier result by combining induction on dimension with a refined analysis of the Ueno locus and Rémond's boundedness theorem.

Core claim

Let A be an abelian variety over a field F, let f: A → P^n be a morphism finite onto its image, and let Γ be a finite-rank subgroup of A(F). Then for any affine chart A^n ⊆ P^n and any finite X ⊆ f(Γ) ∩ A^n one has E(X) ≪ |X|^2 and |X+X| ≫ |X|^2. The proof proceeds by induction on dim A, using the uniform Mordell-Lang conjecture of Gao-Ge-Kühne together with control of the Ueno locus and Rémond's theorem on abelian subvarieties of bounded degree.

What carries the argument

Induction on dimension of A, combined with the uniform Mordell-Lang conjecture, the Ueno locus, and Rémond's boundedness theorem.

If this is right

Images of finite-rank points cannot contain long arithmetic progressions inside affine space.
The result holds for every abelian variety, not only simple ones.
Finite subsets of the image always have large doubling.
The additive structure is controlled uniformly once the morphism is finite onto its image.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the uniform Mordell-Lang conjecture holds, the same rigidity statements may extend to images under rational maps or to points of bounded height.
Direct computational verification for low-dimensional abelian varieties and small-rank subgroups could provide evidence or counter-examples.
The technique may connect to questions about additive bases or sum-product phenomena for other classes of Diophantine sets.

Load-bearing premise

The uniform Mordell-Lang conjecture of Gao-Ge-Kühne holds, the morphism is finite onto its image, and Rémond's boundedness theorem applies to the abelian subvarieties of bounded degree that arise.

What would settle it

An explicit finite-rank subgroup Γ whose image in some affine chart contains a subset X with |X+X| much smaller than |X|^2, such as a long arithmetic progression, would falsify the claim.

read the original abstract

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to a projective space. Let $A/F$ be an 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 $\lvert X+X \rvert \gg \lvert X \rvert^2$. Thus images of finite-rank subgroups of abelian varieties cannot have strong additive structure in affine space. This removes the simplicity assumption from the author's previous result. The proof combines the uniform Mordell--Lang conjecture of Gao--Ge--K\"{u}hne with a refined use of the Ueno locus, R\'{e}mond's boundedness theorem for abelian subvarieties of bounded degree, and induction on the dimension of $A$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the prior result to general abelian varieties but the bounds are conditional on the open uniform Mordell-Lang conjecture.

read the letter

The main takeaway is that this paper removes the simplicity assumption on the abelian variety from the author's earlier work and handles the general case. It does so by running induction on dimension, using the Ueno locus to control bad sets, applying Rémond's boundedness theorem, and invoking the uniform Mordell-Lang conjecture of Gao-Ge-Kühne to get the non-density statement that blocks strong additive structure in the image points.

The argument structure looks coherent on the outline given. The conjecture supplies the finiteness needed so that finite subsets X of the image cannot sit in a coset of a proper subgroup, which then yields the energy upper bound E(X) ≪ |X|^2 and the sumset lower bound |X+X| ≫ |X|^2 inside any affine chart. The dependencies on external theorems appear independent rather than circular.

The clear limitation is that both main claims are conditional. If the uniform Mordell-Lang conjecture fails in some cases, the induction does not close and the rigidity statements may not hold. The abstract states the dependence explicitly, which is helpful, but the result is not unconditional arithmetic geometry. The full derivation is not reproduced here, so any hidden gaps in how the cited theorems are applied cannot be ruled out from the abstract alone.

This is for people already working at the overlap of abelian varieties and additive combinatorics. A reader tracking progress on Mordell-Lang type statements or rigidity phenomena would see a genuine extension of the previous paper and could check the induction step themselves.

It deserves peer review because the extension is non-routine and the proof skeleton is laid out clearly enough to be checked. Recommendation: send it out, with referees asked to focus on whether the conjecture is applied correctly in the induction.

Referee Report

0 major / 1 minor

Summary. The manuscript claims that, assuming the uniform Mordell--Lang conjecture of Gao--Ge--Kühne, if A/F is an abelian variety, f:A→ℙ^n is a morphism finite onto its image, and Γ⊆A(F) is a finite-rank subgroup, then for any affine chart A^n⊆ℙ^n and any finite X⊆f(Γ)∩A^n one has E(X)≪|X|^2 and |X+X|≫|X|^2. The argument proceeds by induction on dim(A), employing a refined analysis of the Ueno locus together with Rémond's boundedness theorem for abelian subvarieties of bounded degree; this removes the simplicity hypothesis present in the author's earlier work.

Significance. Conditional on the Gao--Ge--Kühne conjecture, the result supplies a general statement of additive rigidity for images of finite-rank subgroups under finite morphisms from abelian varieties. It meaningfully extends the previous special-case theorem by eliminating the simplicity assumption and by organizing the proof via dimension induction. The explicit reliance on an external conjecture, the Ueno locus, and Rémond's theorem is clearly acknowledged and constitutes a strength of the manuscript.

minor comments (1)

The energy functional E(X) and the implied constants in the ≪ and ≫ notation are used in the abstract without an immediate definition or reference to the precise dependence (e.g., on A, f, or the rank of Γ); a brief clarifying sentence in the introduction would aid readers outside additive combinatorics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript. The recommendation of minor revision is noted; however, the report contains no specific major comments or requested changes. We therefore see no need for revisions at this stage but remain available to incorporate any further suggestions.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external conjecture and theorems

full rationale

The paper's central claims on energy and sumset bounds are derived by combining the external uniform Mordell--Lang conjecture of Gao--Ge--Kühne, Rémond's boundedness theorem, the Ueno locus, finiteness of the morphism, and induction on dimension. The reference to the author's prior result is only to note removal of a simplicity assumption in the special case; the general-case argument does not reduce to that prior result by definition or construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the provided abstract and outline. The result is explicitly conditional on independent external input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the uniform Mordell--Lang conjecture (not proven in full generality) and standard background results in arithmetic geometry.

axioms (1)

domain assumption Uniform Mordell--Lang conjecture of Gao--Ge--Kühne
Invoked to control the intersection of the image with subvarieties or to bound rational points in the proof strategy described in the abstract.

pith-pipeline@v0.9.1-grok · 5748 in / 1233 out tokens · 29373 ms · 2026-06-28T16:18:18.131496+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 3 canonical work pages · 1 internal anchor

[1]

F. A. Bogomolov, Points of finite order on an abelian variety. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782–804, 973

1980
[2]

Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case

S. Choi, Additive rigidity for images of rational points on abelian varieties, preprint, arxiv.org/abs/2603.24340

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Z. Gao, T. Ge, L. K¨ uhne, The uniform Mordell-Lang conjecture, preprint, arxiv.org/abs/2105.15085

work page arXiv
[4]

Harrison, A

J. Harrison, A. Mudgal, H. Schmidt, Uniform sum-product phenomenon for algebraic groups and Bremner’s conjecture, preprint, arxiv.org/abs/2603.06483

work page arXiv
[5]

Kawamata, On Bloch’s conjecture

Y. Kawamata, On Bloch’s conjecture. Invent. Math. 57 (1980), no. 1, 97–100

1980
[6]

R´ emond, D´ ecompte dans une conjecture de Lang

G. R´ emond, D´ ecompte dans une conjecture de Lang. Invent. Math. 142 (2000), no. 3, 513–545. 12 SEOKHYUN CHOI Dept. of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea Email address:sh021217@kaist.ac.kr

2000

[1] [1]

F. A. Bogomolov, Points of finite order on an abelian variety. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782–804, 973

1980

[2] [2]

Additive Rigidity for Images of Rational Points on Abelian Varieties I: The Simple Case

S. Choi, Additive rigidity for images of rational points on abelian varieties, preprint, arxiv.org/abs/2603.24340

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

Z. Gao, T. Ge, L. K¨ uhne, The uniform Mordell-Lang conjecture, preprint, arxiv.org/abs/2105.15085

work page arXiv

[4] [4]

Harrison, A

J. Harrison, A. Mudgal, H. Schmidt, Uniform sum-product phenomenon for algebraic groups and Bremner’s conjecture, preprint, arxiv.org/abs/2603.06483

work page arXiv

[5] [5]

Kawamata, On Bloch’s conjecture

Y. Kawamata, On Bloch’s conjecture. Invent. Math. 57 (1980), no. 1, 97–100

1980

[6] [6]

R´ emond, D´ ecompte dans une conjecture de Lang

G. R´ emond, D´ ecompte dans une conjecture de Lang. Invent. Math. 142 (2000), no. 3, 513–545. 12 SEOKHYUN CHOI Dept. of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea Email address:sh021217@kaist.ac.kr

2000