Abstract: It has been known for a long time that some genus 1 curves such as
2 y^2 = 1 - 17 x^4
violate the Hasse principle; i.e., they have points over every completion of Q, but not over Q itself. We prove that there exist families of genus 1 curves involving one rational parameter t, such that *every* curve in the family violates the Hasse principle. If more generally, we consider torsors of abelian varieties, then we can find a family for which any value of t defined over an odd degree number field gives a torsor violating the Hasse principle. This is joint work with Jean-Louis Colliot-Thelene.