Palindromic Properties and Descent Obstructions

Speaker: Jim Stankewicz, University of Copenhagen.

For most algebraic curves that you might think of, it is possible to find a twist which has a rational point. For the first time we exhibit an infinite collection of curves over the rational numbers for which this explicitly does not apply. Our family of examples is given by certain Shimura curves, or quotients of the complex upper half-plane which have special properties. Using this family of curves, we find a related set of twists of Shimura curves which all violate the Hasse Principle. This violation is explicitly given by a descent obstruction.