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

2026/03/31 by Seokhyun Choi · 1 citation
Mathematics · #math.NT #math.AG

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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 → ℙⁿ be a morphism which is finite onto its image, and Γ⊆ A(F) be a finite-rank subgroup. We show that for any affine chart \mathbbAⁿ ⊆ ℙⁿ and any finite subset X ⊆ f(Γ) ∩ \mathbbAⁿ, the energy satisfies E(X) ≪ | X |² and the sumset satisfies | X+X | ≫ | X |². We also prove a product version of the main theorem, where the morphism is compatible with the decomposition of the abelian variety into simple factors. The proof uses the uniform Mordell-Lang conjecture proven by Gao--Ge--Kühne.

Citations

• Additive Rigidity for x-Coordinates of Rational Points on Elliptic Curves
• Linear x-coordinate relations of triples on elliptic curves
• On multiplicative energy of subsets of varieties
• Uniformity in Mordell-Lang for curves

Cited by

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

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