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Uniform boundedness of small points on abelian varieties over function fields

Nicole Looper, Jit Wu Yap

Abstract

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that any $K$-rational torsion point $x$ of $A$ has order uniformly bounded in terms of $g$ and the gonality of $B$. We also prove a uniform lower bound on the Néron-Tate height $\widehat{h}_{A,L}(x)$ in terms of the stable Faltings height $h_{\mathrm{Fal}}(A)$ for any $K$-rational point $x$ whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic $0$.

Uniform boundedness of small points on abelian varieties over function fields

Abstract

Let be a field of characteristic and let be the function field of a geometrically irreducible projective curve over . Let be a -dimensional abelian variety with . We prove that any -rational torsion point of has order uniformly bounded in terms of and the gonality of . We also prove a uniform lower bound on the Néron-Tate height in terms of the stable Faltings height for any -rational point whose forward orbit is Zariski dense, proving the Lang-Silverman conjecture over function fields of characteristic .
Paper Structure (21 sections, 74 theorems, 356 equations)

This paper contains 21 sections, 74 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. Proof Strategy
  3. Higher Dimensional Arakelov-Green's Functions
  4. Degeneration by Ultrafilters
  5. Acknowledgements
  6. Background
  7. Heights over Function Fields
  8. Abelian Varieties and Schemes
  9. Moduli Space of Curves
  10. Lower Bound on Arakelov-Green's functions
  11. A General Lower Bound
  12. A Positive Lower Bound for Abelian Varieties
  13. Degenerations of Sequences of Points
  14. Degeneration via Ultrafilters
  15. Application to Moduli of Abelian Varieties
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $K = k(B)$ be the function field of a curve $B/k$, where $\operatorname{char} k=0$. There exists an integer $N = N(g,\operatorname{gon}(B))$, depending only on $g$ and $\operatorname{gon}(B)$, such that for any abelian variety $A/K$ of dimension $g$ with no $K$-isotrivial part, any torsion point

Theorems & Definitions (141)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4: Hindry--Pacheco HP16HP22
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 131 more