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Infinite extensions with finitely many CM moduli

Shu Kawaguchi, Fabien Pazuki

Abstract

We show that there are uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension $g\geqslant1$, generalizing a result of Hultberg in dimension 1.

Infinite extensions with finitely many CM moduli

Abstract

We show that there are uncountably many algebraic extensions of containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension , generalizing a result of Hultberg in dimension 1.
Paper Structure (7 sections, 7 theorems, 25 equations)

This paper contains 7 sections, 7 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Weil height and Northcott numbers
  4. Faltings height
  5. Theta height of level 2
  6. Height and degree inequalities
  7. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.1

Let $g\geqslant1$ be an integer. There exists uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple and principally polarized abelian varieties of dimension $g$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 2.1: Northcott number
  • Definition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 3 more