Abelian threefolds with imaginary multiplication
Francesc Fité, Pip Goodman
TL;DR
This paper analyzes abelian threefolds $A/K$ that admit potential multiplication by an imaginary quadratic field $M$ with signature $(2,1)$ and controlled descent of the endomorphism algebra. By attaching to $A$ a CM elliptic curve $E$ over $K$ via a Galois- and Hecke-theoretic framework, the authors relate the determinant of the $3$-fold wedge representation to the Galois representation of $E$, yielding a bound on the class number $h_M le [L:b Q]/2$ (and, in cases where $ ext{End}^0(A_{ar K})$ is a field, $h_M le [KM:M]$). They provide two complementary proofs: a lambda-adic/Casselman approach for the primary case and a descent argument (with CM-he kernels) for the quadratic extension case, along with a cohomological interpretation in terms of $H^3_{ ext{et}}(A_{ar K},b Q_ ext{ℓ}(1))$. In the special case $K=b Q$, the argument can be completed via modular forms (Eichler–Shimura) and explicit genus $3$ curves with CM endomorphisms are exhibited, illustrating the CM phenomena and bounding the possible imaginary quadratic endomorphism fields. The results illuminate how CM factors control the endomorphism algebra and its arithmetic, with potential applications to zeta-function computations and finite-ness questions for endomorphism algebras.
Abstract
Let A be an abelian threefold defined over a number field K with potential multiplication by an imaginary quadratic field M. If A has signature (2,1) and the multiplication by M is defined over an at most quadratic extension, 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 lambda-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:Q].