Twisting Modular Curves
Ekin Ozman, UT Austin

Abstract: In this talk, we will give results on the certain twists of the classical modular curve. These twists are moduli spaces for a special class of elliptic curves, called Q-curves. Q-curves are interesting for many reasons such as their isogeny class is defined over Q. We will give necessary and sufficient conditions about the existence of local points on these twists. If time permits, we will give examples of curves which have local points everywhere but no points over Q and show that this obstruction to the Hasse Principle is explained by the Brauer-Manin obstruction.