The Brauer Group of Stacky Y_0(2) 

The Brauer groups Br(-) = H^2(-, Gm) of fields and schemes have found much utility in number theory, but Brauer groups of stacks have yet to receive as much attention. In this setting, new phenomena emerge; for example, the Brauer group of a stacky curve over an algebraically closed field can still be non-trivial, i.e. the analogue of Tsen's theorem fails. In this talk, after summarizing some of the previous work on the Gm-cohomology of modular curves, I will explain how one can compute the Brauer group of Y_0(2), the modular curve parameterizing elliptic curves equipped with a cyclic subgroup of order 2. Along the way, I hope to indicate which features of this computation one should expect to generalize to other modular curves as well as to indicate which features I personally am not yet sure how to fit into a nice general picture. This is joint work with Deewang Bhamidipati, Aashraya Jha, Caleb Ji, and Rose Lopez.