Abelian threefolds with imaginary multiplication

Francesc FitÃ©; Pip Goodman

doi:10.1112/jlms.70591

number theory

Abelian threefolds with imaginary multiplication

arXiv:2504.03860 · doi:10.1112/jlms.70591

summary

The paper investigates abelian threefolds over number fields that admit multiplication by an imaginary quadratic field, constructs a related elliptic curve with potential complex multiplication, and shows that the class number of the endomorphism field is bounded by the degree of the compositum of the base field and the quadratic field.

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 $K M$ , 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 $I^»$ -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 $[K M : M]$ .

20 pages. v2: Added section on cohomological interpretation and minor corrections. v3: Final version. Published in JLMS. Missing assumption in Theorem 4.5 (and hence Theorem 1.1) added

Topics & keywords

#abelian varieties #imaginary quadratic multiplication #galois representations #hecke characters #complex multiplication #class number boundsabelian threefoldimaginary quadratic fieldpotential multiplicationλ-adic representationHecke characterclass number bound