Additive Rigidity for Images of Rational Points on Abelian Varieties

Seokhyun Choi

arxiv: 2603.24340 · v2 · submitted 2026-03-25 · 🧮 math.NT · math.AG

Additive Rigidity for Images of Rational Points on Abelian Varieties

Seokhyun Choi This is my paper

Pith reviewed 2026-05-15 00:34 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords abelian varietiesMordell-Lang conjectureadditive energysumset sizerational pointsDiophantine geometryadditive combinatorics

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The pith

Finite morphisms from abelian varieties to projective space make images of finite-rank rational point groups additively rigid.

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

The paper proves that when a morphism f from a simple abelian variety A to projective space is finite onto its image, the points X drawn from f(Γ) inside any affine chart A^n have additive energy E(X) at most a constant times |X| squared and sumset size at least a constant times |X| squared. This means the image points avoid most additive relations and behave like generic sets under addition. The same conclusion is shown to hold for non-simple abelian varieties once f respects the decomposition into simple factors. The argument relies on the uniform Mordell-Lang conjecture to limit how the preimages can intersect subvarieties that would create additive structure.

Core 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.

What carries the argument

Finite morphism f from the abelian variety onto its image in projective space, combined with the uniform Mordell-Lang conjecture applied to preimages to control intersections that would produce additive relations.

If this is right

The image points X satisfy few solutions to equations of the form x + y = z + w with distinct pairs.
The points cannot lie in low-dimensional additive subgroups or contain long arithmetic progressions in the affine chart.
The rigidity persists when the abelian variety decomposes into simple factors provided the morphism respects that decomposition.
The result supplies a uniform arithmetic constraint on how rational points from finite-rank groups can accumulate in affine space.

Where Pith is reading between the lines

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

The same method may yield effective bounds on the number of points of bounded height inside certain regions of the affine chart.
Direct computation on elliptic curves with known large finite-rank subgroups could produce numerical checks on the implied constants.
The technique suggests possible links between Mordell-Lang statements and sumset estimates in other Diophantine settings.
It is natural to ask whether the finiteness assumption on f can be relaxed while keeping the rigidity conclusion.

Load-bearing premise

The uniform Mordell-Lang conjecture holds for the abelian varieties and the subvarieties cut out by the affine charts and the morphism.

What would settle it

An explicit example of a simple abelian variety, a finite morphism f, and arbitrarily large finite X inside f(Γ) ∩ A^n with |X+X| much smaller than |X|^2 would disprove the rigidity 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 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 the same additive rigidity holds for arbitrary abelian varieties, and prove that this is indeed the case when the morphism $f$ is compatible with the decomposition of $A$ into simple factors. The proof uses the uniform Mordell-Lang conjecture.

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 paper gives conditional additive rigidity bounds E(X) ≪ |X|^2 and |X+X| ≫ |X|^2 for images of finite-rank rational points on abelian varieties under morphisms to projective space, relying on the uniform Mordell-Lang conjecture.

read the letter

The core point is that the paper proves these additive-combinatorial bounds for the images f(Γ) inside affine charts, but only when the uniform Mordell-Lang conjecture is assumed. For a simple abelian variety A and a finite morphism f onto its image, any finite X inside f(Γ) ∩ A^n satisfies the energy bound and the sumset lower bound. The argument then extends to general A by requiring that f respects the decomposition into simple factors. This is the new piece: applying sumset and energy estimates directly to these arithmetic images rather than to arbitrary finite sets in projective space. The outline is consistent with standard uses of Mordell-Lang to limit intersections with subvarieties, and the finite-morphism hypothesis is stated clearly up front. Credit is due for keeping the reduction step explicit instead of burying it. The main limitation is the dependence on the external conjecture; without it the claims are empty, and the paper does not offer unconditional corollaries or special cases where the conjecture can be avoided. The compatibility condition for non-simple varieties also narrows the scope, though that is flagged rather than hidden. I only saw the abstract and outline, so I cannot check the exact derivation steps, but nothing in the stated logic looks circular or internally inconsistent. This is for arithmetic geometers who already work with rational points on abelian varieties and want to see how additive combinatorics can be brought in under known conjectures. A reader who follows Mordell-Lang applications or cares about distribution of points in affine space will get something concrete from it. It is worth sending to referees so the details can be verified and the conditional framing can be tightened if needed.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes additive rigidity results for the images of finite-rank subgroups of rational points on abelian varieties under morphisms to projective space. For simple abelian varieties A/F with finite morphism f to P^n, and finite-rank subgroup Γ, any finite X ⊆ f(Γ) ∩ A^n satisfies E(X) ≪ |X|^2 and |X+X| ≫ |X|^2. The result extends to general abelian varieties when f is compatible with the decomposition into simple factors. The proofs rely on the uniform Mordell-Lang conjecture.

Significance. Conditional on the uniform Mordell-Lang conjecture, the results show that such images of rational points exhibit strong additive rigidity, behaving like generic sets in additive combinatorics. This provides a precise bridge between Diophantine geometry and sumset theory, with clear hypotheses on f (finiteness and compatibility) that make the claims falsifiable in special cases. The explicit conditional framing and reduction via simple factors are strengths.

major comments (2)

[Introduction and main theorems] The reduction step for general abelian varieties (extending the simple case) requires the morphism f to be compatible with the simple-factor decomposition; this condition is load-bearing for the general statement but is only defined after the simple-case theorem, so the introduction should state it explicitly alongside the main claims.
[Proof of the simple case] The application of the uniform Mordell-Lang conjecture to bound intersections with subvarieties in the affine chart needs a precise reference to how finiteness of f ensures the relevant subvarieties are proper; without this, the energy and sumset bounds in the simple case rest on an implicit step.

minor comments (2)

[Notation and statements] The implicit constants in the ≪ and ≫ notations should be stated to depend only on A, f, and the rank of Γ (not on the choice of X).
[Preliminaries] Add a reference to a standard formulation of the uniform Mordell-Lang conjecture in the introduction or preliminaries.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below.

read point-by-point responses

Referee: [Introduction and main theorems] The reduction step for general abelian varieties (extending the simple case) requires the morphism f to be compatible with the simple-factor decomposition; this condition is load-bearing for the general statement but is only defined after the simple-case theorem, so the introduction should state it explicitly alongside the main claims.

Authors: We agree that the compatibility condition is essential for the general statement and should be stated explicitly in the introduction. In the revised manuscript we will move the definition of compatibility forward and restate the general theorem in the introduction with this hypothesis included alongside the simple-case result. revision: yes
Referee: [Proof of the simple case] The application of the uniform Mordell-Lang conjecture to bound intersections with subvarieties in the affine chart needs a precise reference to how finiteness of f ensures the relevant subvarieties are proper; without this, the energy and sumset bounds in the simple case rest on an implicit step.

Authors: We accept that the argument that finiteness of f implies the relevant subvarieties are proper is currently implicit. In the revision we will insert an explicit sentence in the proof of the simple case, citing the fact that a finite morphism pulls back proper subvarieties of the image to proper subvarieties of A, so that the uniform Mordell-Lang conjecture applies directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; result is conditional on external conjecture

full rationale

The paper derives the energy bound E(X) ≪ |X|^2 and sumset lower bound |X+X| ≫ |X|^2 for finite subsets X of f(Γ) ∩ A^n by invoking the uniform Mordell-Lang conjecture as an external hypothesis, together with the finiteness of f onto its image and compatibility with simple-factor decomposition in the general case. No step reduces a claimed prediction to a fitted parameter from the same data, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified within the paper. The argument remains self-contained once the stated conjecture is granted, with no self-definitional loops or ansatz smuggling visible in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the uniform Mordell-Lang conjecture as the key external assumption for the general case; no free parameters or new entities are introduced.

axioms (1)

domain assumption Uniform Mordell-Lang conjecture
Invoked to control the distribution of rational points and thereby obtain the additive estimates for general abelian varieties.

pith-pipeline@v0.9.0 · 5476 in / 1214 out tokens · 40484 ms · 2026-05-15T00:34:42.963312+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 ... E(X) ≤ C(g,d,t)^{1+r} |X|^2 ... uses uniform Mordell-Lang conjecture [6]
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 4.1 ... deg L(Y) ≤ 2^g d^2 t^2 ... apply Theorem 3.1

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