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Explicit Zsigmondy bounds for families of Drinfeld modules of rank 2

Matias Alvarado

TL;DR

This work establishes explicit Zsigmondy bounds for families of rank-2 Drinfeld modules over function fields by bounding local and canonical heights. The authors show that, for $\phi_T(x)=Tx+gx^q+\Delta x^{q^2}$ with suitable coprimality and splitting conditions, the maximal index in the Zsigmondy set $\mathcal{Z}(\phi,x)$ is bounded by an explicit function of $N$, the set of places $S$, and the degrees of $g$ and $\Delta$, namely $\max\mathcal{Z}(\phi,x) \le (6(q^{2N}-1)N|S|+4)+\tfrac{1}{2}\log_q(2+2\deg g+2\deg \Delta)$. The method combines precise control of local heights, a comparison between the classical height $h$ and the canonical height $\hat{h}$, and a reduction to the case where the discriminant splits via a base change to $\mathbb{F}_{q^N}$. The result extends finiteness theorems to effective, explicit bounds and provides a toolkit for effective arithmetic dynamics of Drinfeld modules, with potential applications to decidability questions akin to elliptic divisibility sequences in the function-field setting.

Abstract

We give explicit bounds for Zsigmondy sets of certain families of Drinfeld modules of rank 2. The primary strategy is to bound the local heights associated to Drinfeld modules and then relate canonical to classical heights.

Explicit Zsigmondy bounds for families of Drinfeld modules of rank 2

TL;DR

This work establishes explicit Zsigmondy bounds for families of rank-2 Drinfeld modules over function fields by bounding local and canonical heights. The authors show that, for with suitable coprimality and splitting conditions, the maximal index in the Zsigmondy set is bounded by an explicit function of , the set of places , and the degrees of and , namely . The method combines precise control of local heights, a comparison between the classical height and the canonical height , and a reduction to the case where the discriminant splits via a base change to . The result extends finiteness theorems to effective, explicit bounds and provides a toolkit for effective arithmetic dynamics of Drinfeld modules, with potential applications to decidability questions akin to elliptic divisibility sequences in the function-field setting.

Abstract

We give explicit bounds for Zsigmondy sets of certain families of Drinfeld modules of rank 2. The primary strategy is to bound the local heights associated to Drinfeld modules and then relate canonical to classical heights.
Paper Structure (10 sections, 11 theorems, 58 equations)

This paper contains 10 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Heights
  3. Heights and Drinfeld modules
  4. Difference between $h$ and $\hat{h}$
  5. Bounds in rank 2
  6. Previous results and strategy
  7. Local heights
  8. Estimates
  9. Bound for $\hat{h}(x)$
  10. Proof of the main theorem

Key Result

Theorem 1.1

Let $\phi \colon \mathbb{F}_q[T] \to \mathrm{End}_{\mathbb{F}_{q}(T)}(\mathbb{G}_a)$ be a Drinfeld module of rank $2$ given by where $g,\Delta \in \mathbb{F}_q[T]$. Let $N$ be the least common multiple of the degrees of irreducible divisors of $\Delta$. Let us suppose that $\phi_T(1)$ is coprime to $T\Delta$ in $\mathbb{F}_q[T].$ Let $S$ be the finite set of places $\{v \in M_{\mathbb{F}_q(T)}:v\

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 1
  • Remark 2.1
  • Definition 2: Denis, denis
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 17 more