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Manin-Mumford in arithmetic pencils

Gregorio Baldi, Rodolphe Richard, Emmanuel Ullmo

Abstract

We obtain a refinement of Manin-Mumford (Raynaud's Theorem) for abelian schemes over some ring of integers. Torsion points are replaced by special 0-cycles, that is reductions modulo some, possibly varying, prime of Galois orbits of torsion points. There is a flat/horizontal part and a vertical one. The irreducible components of the flat part are given by the Zariski closure, over the integers, of torsion cosets of the generic fibre of the abelian scheme. The vertical components are given by translates of abelian subvarieties, which 'come from characteristic zero'.

Manin-Mumford in arithmetic pencils

Abstract

We obtain a refinement of Manin-Mumford (Raynaud's Theorem) for abelian schemes over some ring of integers. Torsion points are replaced by special 0-cycles, that is reductions modulo some, possibly varying, prime of Galois orbits of torsion points. There is a flat/horizontal part and a vertical one. The irreducible components of the flat part are given by the Zariski closure, over the integers, of torsion cosets of the generic fibre of the abelian scheme. The vertical components are given by translates of abelian subvarieties, which 'come from characteristic zero'.

Paper Structure

This paper contains 49 sections, 38 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Strategy of the proof and outline of the paper
  5. Horizontal and weak vertical
  6. Vertical and lifting
  7. Points of $p$-torsion in characteristic $p$
  8. Preliminaries and Hindry's method
  9. A first reduction
  10. Lang group and adelic Tate module
  11. Degree, stabiliser subgroup, and a criterion for specialness
  12. Horizontal and weak vertical, Theorem \ref{['mainthm']}
  13. Statement of the main proposition
  14. Proof of Proposition \ref{['keyprop']}
  15. First lemma
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $A$ be an abelian variety over a field of $k$ characteristic $0$ and $E$ be a subset of torsion points of $A(\overline{k})$. Then the Zariski closure of $E$ is special: a finite union of torsion cosets $a+B$ for some abelian subvariety $B\subseteq A_{\overline{k}}$ and a torsion point $a\in A(\o

Theorems & Definitions (73)

  • Theorem 1.1: Raynaud
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 63 more