As E varies among elliptic curves defined over the rational numbers, a theorem of Bhargava and Shankar shows that the average rank of the Mordell--Weil group E(Q) is bounded. If we now fix a number field K, is the same true of E(K)? I will report on recent progress on this question, answering this question in the affirmative for certain choices of K. This progress follows from a study of certain local invariants of elliptic curves, which loosely describe the failure of Galois descent for the associated p-Selmer groups.