Vol. 14, No. 9, 2020

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The Brauer group of the moduli stack of elliptic curves

Benjamin Antieau and Lennart Meier

Vol. 14 (2020), No. 9, 2295–2333
Abstract

We compute the Brauer group of ${\mathsc{ℳ}}_{1,1}$, the moduli stack of elliptic curves, over $Specℤ$, 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 $\mathsc{ℳ}\left(2\right)$ 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 ${S}_{3}$ 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.

Keywords
Brauer groups, moduli of elliptic curves, level structures, Hilbert symbols
Mathematical Subject Classification 2010
Primary: 14F22
Secondary: 14H52, 14K10
Milestones
Received: 10 November 2017
Revised: 26 March 2020
Accepted: 4 May 2020
Published: 13 October 2020
Authors
 Benjamin Antieau Department of Mathematics Northwestern University Chicago, IL United States Lennart Meier Mathematical Institut Universiteit Utrecht Utrecht Netherlands