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

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, $f:A\to {\mathbb{P}}^n$ 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 $E(X)\ll |X{|}^2$ and the sumset satisfies $|X+X|\gg |X{|}^2$. 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.

18 pages