Additive Rigidity for Images of Rational Points on Abelian Varieties II: The General 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 a projective space. Let $A/F$ be an 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 $|X+X|\gg |X{|}^2$. 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--Kühne with a refined use of the Ueno locus, Rémond's boundedness theorem for abelian subvarieties of bounded degree, and induction on the dimension of $A$.

12 pages