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Heights of Drinfeld modular polynomials and Hecke images

Florian Breuer, Fabien Pazuki, Zhenlin Ran

Abstract

We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials $Φ_N$ for any monic $N\in\mathbb{F}_q[t]$. These polynomials vanish at pairs of $j$-invariants of Drinfeld $\mathbb{F}_q[t]$-modules of rank 2 linked by cyclic isogenies of degree $N$. The main term in both bounds is asymptotically optimal as $\mathrm{deg}(N)$ tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2.

Heights of Drinfeld modular polynomials and Hecke images

Abstract

We obtain explicit upper and lower bounds on the size of the coefficients of the Drinfeld modular polynomials for any monic . These polynomials vanish at pairs of -invariants of Drinfeld -modules of rank 2 linked by cyclic isogenies of degree . The main term in both bounds is asymptotically optimal as tends to infinity. We also obtain precise estimates on the Weil height and Taguchi height of Hecke images of Drinfeld modules of rank 2.

Paper Structure

This paper contains 16 sections, 26 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Heights
  4. Lattices and the Drinfeld upper half plane
  5. Drinfeld modular forms
  6. Bruhat-Tits trees
  7. The Building map
  8. Farey sequences
  9. Size of representatives
  10. Proof of Theorem \ref{['thm:main']}
  11. The case $|d_\gamma| \leq \sqrt{|N||z|_{\mathrm{i}}}$
  12. The case $|d_\gamma|>\sqrt{|N||z|_\mathrm{i}}$
  13. Specialization
  14. Heights of Hecke images
  15. Proof of Theorem \ref{['Silverman']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $N\in A$ be monic. Then the height of the modular polynomial $\Phi_N(X,Y)$ satisfies where

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 42 more