Additive decompositions for rings of modular forms

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen--Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.Comment: Rewritten introduction, updated references. This article supersedes the algebraic part of arXiv:1609.0926

Fibration Categories are Fibrant Relative Categories

A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick--Kan model structure.Comment: 21 pages; comments welcom

Fibrancy of Partial Model Categories

We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category is fibrant. Furthermore, we describe connections to calculi of fractions.Comment: 30 page

A Whitehead theorem for periodic homotopy groups

We show that $v_n$-periodic homotopy groups detect homotopy equivalences between simply-connected finite CW-complexes

Connective Models for Topological Modular Forms of Level nn

The goal of this article is to construct and study connective versions of topological modular forms of higher level like $\mathrm{tmf}_1(n)$. In particular, we use them to realize Hirzebruch's level-$n$ genus as a map of ring spectra.Comment: 27 pages; v2: added several clarifications and minor correcionts in response to referee's comments, final version to appear in AG

United Elliptic Homology

We study the categories of modules over real K-theory and TMF. Inspired by work of Bousfield, we consider TMF-modules at the prime 3 which are relatively free with respect to TMF(2). We show that a large class of these can be iteratively built from TMF by coning off torsion elements and killing generators. This is based on a detailed study of vector bundles on the moduli stack of elliptic curves. Furthermore, we consider examples of TMF-modules and show that the categories of TMF-modules and quasi-coherent sheaves on the derived moduli stack of elliptic curves are equivalent (at primes bigger than 2)