The Brauer group of the moduli stack of elliptic curves

Mumford proved that the Picard group of the moduli stack of elliptic curves is a finite group of order 12, generated by the Hodge bundle of the universal family of elliptic curves. I will talk about new work with Lennart Meier, motivated by chromatic homotopy theory, which considers the Brauer group of the moduli stack and shows that it is zero. To do so, we compute the Brauer group of the moduli stack of elliptic curves with full level 2 structure, which is a 2-group of order 2, and we find examples of elliptic curves over the 2-adic integers that exhibit non-zero ramification for each class. Little background will be assumed.