University of Sydney Algebra Seminar

Francesc Fité

Friday 8 August, 12-1pm, in SMRI Seminar Room (Macleay Building A12 Room 301)

Elliptic curves attached to abelian threefolds with imaginary multiplication

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 \(\lambda\)-adic representations attached to \(A\). This exhibits an \(8\)-dimensional subspace of the degree \(3\) cohomology of \(A\) coming from an abelian variety, namely \(A \times E\). 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 \(-1\).