The cohomology theory known as (Formula presented.), 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 (Formula presented.) with level structure to forms of (Formula presented.)-theory. In particular, this allows us to construct a connective spectrum (Formula presented.) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces (Formula presented.) with level structure.
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The authors would like to thank Matthew Ando, Mark Behrens, Andrew Blumberg, Scott Carnahan, Jordan Ellenberg, Paul Goerss, Mike Hopkins, Nitu Kitchloo, Michael Mandell, Akhil Mathew, Lennart Meier, Niko Naumann, William Messing, Arthur Ogus, Kyle Ormsby, Charles Rezk, Andrew Salch, George Schaeffer, and Vesna Stojanoska for discussions related to this paper. The anonymous referee of [31 ] also asked a critical question about compatibility with Z / 2-actions, motivating our proof that evaluation at the cusp is possible. The ideas in this paper would not have existed without the Loen conference "p-Adic Geometry and Homotopy Theory" introducing us to logarithmic structures in 2009; the authors would like to thank the participants there, as well as Clark Barwick and John Rognes for organizing it. This paper is written in dedication to Mark Mahowald. M. Hill was partially supported by NSF DMS?0906285, DARPA FA9550?07?1?0555, and the Sloan foundation. Tyler Lawson was partially supported by NSF DMS?1206008 and the Sloan foundation.
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