Construction of Hecke Characters for Three-dimensional CM Abelian Varieties

TL;DR

This paper confirms the exact list of 37 CM fields for three-dimensional CM abelian varieties with rational moduli and constructs Hecke characters to determine which classes have rational models, revealing a deviation from the elliptic and surface cases.

Contribution

It precisely identifies the 37 CM fields and constructs Hecke characters to determine which abelian varieties have rational models, extending previous results to three dimensions.

Findings

Exact list of 37 CM fields for three-dimensional CM abelian varieties.

28 isogeny classes have $ ext{Q}$-models.

Construction of Hecke characters satisfying Shimura's theorem.

Abstract

It is well-known for an elliptic curve with complex multiplication that the existence of a $\mathbb{Q}$-rational model is equivalent to its field of moduli being equal to $\mathbb{Q}$, or its endomorphism ring being the ring of integers of 9 possible fields ($\ast$). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove that precisely 28 isogeny classes of these abelian varieties have $\mathbb{Q}$-models. Therefore the complete analogy to $(\ast)$ fails here.