Department of

October 2016 November 2016 December 2016 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 4 5 1 2 3 2 3 4 5 6 7 8 6 7 8 9 10 11 12 4 5 6 7 8 9 10 9 10 11 12 13 14 15 13 14 15 16 17 18 19 11 12 13 14 15 16 17 16 17 18 19 20 21 22 20 21 22 23 24 25 26 18 19 20 21 22 23 24 23 24 25 26 27 28 29 27 28 29 30 25 26 27 28 29 30 31 30 31

Tuesday, November 29, 2016

**Abstract:** 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. After giving background on Brauer groups of ring spectra and our motivation, I will talk about recent work with Lennart Meier, motivated by elliptic cohomology, which considers the Brauer group of the moduli stack and shows that it vanishes. Non-6-torsion can be handled by general descent spectral sequence methods. To handle the 2-torsion, we study the moduli stack of elliptic curves with full level 2 structure, showing with geometric and arithmetic arguments that it is a 2-group. Then, by exhibiting 2-adic elliptic curves with certain discriminants, we show that the non-zero classes do not extend to the integral moduli stack.