The Shimura curve of discriminant 15 and topological automorphic forms

We find defining equations for the Shimura curve of discriminant 15 over Z[1/15]. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of "topological automorphic forms" associated to this curve, as well as one associated to a quotient by an Atkin-Lehner involution.Comment: 36 pages, 5 figures, updated with corrections and new introduction (this version corrects image issues in the previous

Structured ring spectra and displays

We combine Lurie's generalization of the Hopkins-Miller theorem with work of Zink-Lau on displays to give a functorial construction of even-periodic commutative ring spectra, concentrated in chromatic layers 2 and above, associated to certain n by n invertible matrices with coefficients in Witt rings. This is applied to examples related to Lubin-Tate and Johnson-Wilson spectra. We also give a Hopf algebroid presentation of the moduli of p-divisible groups of height greater than or equal to 2.Comment: 18 page

The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its "deformation representation ring," analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices

Regularity of structured ring spectra and localization in K-theory

We identify a regularity property for structured ring spectra, and with it we prove a natural analogue of Quillen's localization theorem for algebraic K-theory in this setting.Comment: 9 pages. Corrected references and aspects of the expositio