Topological modular forms with level structure

By M. Hill, Tyler Lawson. Inventiones Mathematicae, 2013

67 citations recorded by the source corpus.

The cohomology theory known as Tmf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Tmf}$$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Tmf}$$\end{document} with level structure to forms of K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K$$\end{document}-theory. In particular, this allows us to construct a connective spectrum tmf0(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{tmf}_0(3)$$\end{document} 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 Tmf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Tmf}$$\end{document} with level structure.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0af3727d8cde4b943f766f7e431a3b9f85477ad3

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In Geometry and Topology

1. Lennart Meier
2. A. Mathew
3. M. Hill
4. B. Bhatt
5. Peter Scholze
6. N. Axel Naumann
7. J. Noel
8. Vesna Stojanoska

In Mathematical Physics

1. A. Mathew
2. Lennart Meier
3. M. Hill
4. B. Bhatt
5. Peter Scholze
6. N. Axel Naumann
7. J. Noel
8. Vesna Stojanoska

Citations in the corpus

• The homology of $\tmf$ (2013, background): “Using similar techniques as in Theorem 4.12 and the Hill–Lawson work [12], one obtains from the above analysis: Theorem 4.15.”
• The C2–spectrum Tmf1(3) and its invertible modules (2015, background): “As a first step, Hill and Lawson overcame in [29] certain technical obstacles to define E1–ring spectra Tmf1.”
• Affineness and chromatic homotopy theory (2013, methodology): “Theorem 7.10 (Goerss–Hopkins and Hill–Lawson [23]).”