TY - JOUR

T1 - The brauer group of the moduli stack of elliptic curves

AU - Antieau, Benjamin

AU - Meier, Lennart

AU - Olsson, Martin

N1 - Funding Information:
Benjamin Antieau was supported by NSF Grants DMS-1461847 and DMS-1552766. Lennart Meier was supported by DFG SPP 1786. We thank the Hausdorff Research Institute for Mathematics, UIC, and Universit?t Bonn for hosting one or the other author in 2015 and 2016 while this paper was being written. Furthermore, we thank Asher Auel, Akhil Mathew, and Vesna Stojanoska for their ideas at the beginning of this project and Peter Scholze and Yichao Tian for the suggestion to use fppf-cohomology and the idea for the splitting in Proposition 2.9. Finally, we thank the anonymous referees for their helpful suggestions.
Funding Information:
Benjamin Antieau was supported by NSF Grants DMS-1461847 and DMS-1552766. Lennart Meier was supported by DFG SPP 1786. MSC2010: primary 14F22; secondary 14H52, 14K10. Keywords: Brauer groups, moduli of elliptic curves, level structures, Hilbert symbols.
Publisher Copyright:
© 2020 Mathematical Sciences Publishers.

PY - 2020

Y1 - 2020

N2 - We compute the Brauer group of M1;1, the moduli stack of elliptic curves, over Spec ℤ, its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not 2. The methods involved include the use of the parameter space of Legendre curves and the moduli stack M(2) of curves with full (naive) level 2 structure, the study of the Leray-Serre spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of S3 in a certain integral representation, the classification of cubic Galois extensions of ℚ, the computation of Hilbert symbols in the ramified case for the primes 2 and 3, and finding p-adic elliptic curves with specified properties.

AB - We compute the Brauer group of M1;1, the moduli stack of elliptic curves, over Spec ℤ, its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not 2. The methods involved include the use of the parameter space of Legendre curves and the moduli stack M(2) of curves with full (naive) level 2 structure, the study of the Leray-Serre spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of S3 in a certain integral representation, the classification of cubic Galois extensions of ℚ, the computation of Hilbert symbols in the ramified case for the primes 2 and 3, and finding p-adic elliptic curves with specified properties.

KW - Brauer groups

KW - Hilbert symbols

KW - Level structures

KW - Moduli of elliptic curves

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U2 - 10.2140/ant.2020.14.2295

DO - 10.2140/ant.2020.14.2295

M3 - Article

AN - SCOPUS:85092940105

VL - 14

SP - 2295

EP - 2333

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 9

ER -