Algebra Seminar: Fite -- Elliptic curves attached to abelian threefolds with imaginary multiplication

Francesc Fite (University of Barcelona) will be speaking in the algebra seminar.  We
will go out for lunch after the talk---all are welcome to join! 

When: 12-1pm Friday August 8 

Where: SMRI Seminar Room (Macleay Building A12 Room 301) 

Title: Elliptic curves attached to abelian threefolds with imaginary multiplication 

Abstract: The classical theory of Shimura and Taniyama attaches to an abelian variety
with complex multiplication defined over a number field an algebraic Hecke character
with infinity type determined by the CM type of the given abelian variety.  A converse
theorem by Casselman attaches a CM abelian variety to an algebraic Hecke character whose
infinity type is of the right form.  Let A be an abelian threefold defined over a number
field K whose geometric endomorphism algebra is an imaginary quadratic field M.  I will
explain a joint result with Pip Goodman that uses Casselman's theorem to attach to A an
elliptic curve E defined over K with potential complex multiplication by M with the
following property: the Hecke character of E coincides with the Tate twisted determinant
of the compatible system of λ-adic representations attached to A.  This exhibits an
8-dimensional subspace of the degree 3 cohomology of A coming from an abelian variety,
namely A x E.  Time permitting, I will report on ongoing work with X.  Guitart and F.
Pedret, where we explicitly determine (the isogeny class of) E when A is either the
Jacobian of a Picard curve (imaginary multiplication by square root of -3) or the
Jacobian of a hyperelliptic genus 3 curve with imaginary multiplication by square root
of -1.