TY - JOUR
T1 - Topological modular forms with level structure
AU - Hill, Michael
AU - Lawson, Tyler
N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
PY - 2016/2/1
Y1 - 2016/2/1
N2 - 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.
AB - 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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U2 - 10.1007/s00222-015-0589-5
DO - 10.1007/s00222-015-0589-5
M3 - Article
AN - SCOPUS:84957849843
SN - 0020-9910
VL - 203
SP - 359
EP - 416
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 2
ER -