Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On abelian points of varieties intersecting subgroups in a torus

Jorge Mello

Abstract

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.

On abelian points of varieties intersecting subgroups in a torus

Abstract

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety intersected with the union of connected algebraic subgroups of codimension at least in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.

Paper Structure

This paper contains 7 sections, 16 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Preliminar definitions and results
  5. Proofs of the main results
  6. A structure theorem over $K_{ab}$
  7. Abelian points in curves and periodic hypersurfaces

Key Result

Theorem 1

OSSZ1Let$X \subset G= \mathbb{G}_m^n$be an irreducible curve defined over a number field$K$and not contained in any translate of a proper algebraic subgroup of $\mathbb{G}_m^n$. Suppose that$L \supset K$has the Bogomolov Property. Then there are at most finitely many points in $X(L)$ whose coordinat

Theorems & Definitions (31)

  • Theorem
  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 21 more