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Heights and singular moduli of Drinfeld modules

Zhenlin Ran

Abstract

Let $q$ be an odd number and $q>5$, and $\mathbb{F}_q$ be a finite field of $q$ elements. We prove that at most finitely many singular moduli of rank 2 $\mathbb{F}_q[t]$-Drinfeld modules are algebraic units. In particular, we develop some techniques of heights of Drinfeld modules to approach it.

Heights and singular moduli of Drinfeld modules

Abstract

Let be an odd number and , and be a finite field of elements. We prove that at most finitely many singular moduli of rank 2 -Drinfeld modules are algebraic units. In particular, we develop some techniques of heights of Drinfeld modules to approach it.
Paper Structure (6 sections, 30 theorems, 97 equations)

This paper contains 6 sections, 30 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Minimal models of Drinfeld $A$-modules
  3. Heights
  4. Arithmetic on quadratic fundamental domain
  5. Bounding $h(J)$
  6. More on Complex Multiplication

Key Result

Theorem 1.1

At most finitely many singular moduli are algebraic units.

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 56 more