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

We compute the Brauer group of 1,1, the moduli stack of elliptic curves, over Spec , 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 (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 , the computation of Hilbert symbols in the ramified case for the primes 2 and 3, and finding p-adic elliptic curves with specified properties.

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