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Additive Rigidity for Images of Rational Points on Abelian Varieties

Seokhyun Choi

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 $A$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $Γ\subseteq A(\overline{K})$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(Γ) \cap \mathbb{A}^n$, the energy and the sumset of $X$ satisfy quadratic bounds in $\lvert X \rvert$. The proof uses the uniform Mordell-Lang conjecture.

Additive Rigidity for Images of Rational Points on Abelian Varieties

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 be a simple abelian variety, be a morphism which is finite onto its image, and be a finite-rank subgroup. We show that for any affine chart and any finite subset , the energy and the sumset of satisfy quadratic bounds in . The proof uses the uniform Mordell-Lang conjecture.

Paper Structure

This paper contains 4 sections, 8 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Reduction of Theorem \ref{['main_theorem']}
  3. Finiteness of $Y_u(F) \cap \Gamma^2$
  4. Finiteness of $\Sigma$

Key Result

Theorem 1.1

Let $A/F$ be a simple abelian variety of dimension $g$ over an algebraically closed field $F$ of characteristic 0. Let $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite of degree $d$ onto its image, and let $t$ denote the projective degree of $f(A)$ in $\mathbb{P}^n$. Let $\Gamma$ be a su

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1: Uniform Mordell-Lang conjecture
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 4 more