Infinite extensions with finitely many CM moduli

TL;DR

This paper demonstrates the existence of uncountably many algebraic extensions of the rational numbers that contain only finitely many moduli of CM simple principally polarized abelian varieties for any fixed dimension, extending previous results in dimension 1.

Contribution

It generalizes Hultberg's result from dimension 1 to higher dimensions, showing the abundance of such algebraic extensions.

Findings

Uncountably many algebraic extensions with finitely many CM moduli exist for any dimension.

Extension of Hultberg's dimension 1 result to higher dimensions.

Finiteness of CM moduli in these extensions is established.

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.