Abelian threefolds with imaginary multiplication

Abstract

Let $A$ be an abelian threefold defined over a number field $K$ with potential multiplication by an imaginary quadratic field $M$. Under mild assumptions on $K$, if $A$ has signature $(2,1)$ and the multiplication by $M$ is defined over $KM$, we attach to $A$ an elliptic curve defined over $K$ with potential complex multiplication by $M$, whose attached Galois representation is determined by the Hecke character associated to the determinant of the compatible system of $λ$-adic representations of $A$. We deduce that if the geometric endomorphism algebra of $A$ is an imaginary quadratic field, then it necessarily has class number bounded by $[KM:M]$.