2026/06/30 by Seokhyun Choi
Mathematics · #math.NT #math.AG #msc:11G05 #msc:11G10
12 pages
arxiv created 2026/08/03 · arxiv updated 2026/08/04
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 → 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. 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.