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

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 $A\mathrm{/}F$ be an abelian variety, $f:A\to {\mathbb{P}}^n$ be a morphism which is finite onto its image, and $\mathrm{\Gamma }\subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart ${\mathbb{A}}^n\subseteq {\mathbb{P}}^n$ and any finite subset $X\subseteq f(\mathrm{\Gamma })\cap {\mathbb{A}}^n$, the energy satisfies $E(X)\ll \mid X{\mid }^2$ and the sumset satisfies $\mid X+X\mid \gg \mid X{\mid }^2$. 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 $A$.