Abelian threefolds with imaginary multiplication, and elliptic curves attached to them

Abstract: It is an interesting problem to construct algebraic curves whose Jacobians have extra endomorphisms. When genus is 3, there are two natural families of Jacobians with imaginary multiplication by $\mathbb{Z}[i]$ and $\mathbb{Z}[\zeta_3]$, coming from curves with a $\mu_4$ or $\mu_6$ action. We will report on new hyperelliptic families with imaginary multiplication by $\mathbb{Z}[\sqrt{-d}]$ for $d=2,3,4$, and some instances of extending to any odd genus g. Galois representations allow one to naturally attach a CM elliptic curve to any abelian threefold with imaginary multiplication of signature (2,1). For the new families, we will explicitly compute the attached CM elliptic curve. An analogue of this association was used by Laga-Shnidman to get results on vanishing of Ceresa cycles for Picard curves. Based on ongoing work with Francesc Fite and Pip Goodman.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar