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Register a new userOn the ring of cooperations for 2-primary connective topological modular forms
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We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.
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We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the building complex associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma_0(3)).
The cohomology theory known as Tmf, for topological modular forms, is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf_0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a sheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-etale site of the moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf with level structure.
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