Intermediate Jacobians and rationality over arbitrary fields

Clemens and Griffiths have used intermediate Jacobians to show that smooth cubic threefolds are irrational. In this talk, I will explain how to extend the Clemens-Griffiths method over arbitrary fields. As an application, I will show that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. This is joint work with Olivier Wittenberg.