The Brauer group of the moduli stack of elliptic curves

Antieau, Benjamin; Meier, Lennart

doi:10.2140/ant.2020.14.2295

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

The Brauer group of the moduli stack of elliptic curves

Benjamin Antieau and Lennart Meier

Vol. 14 (2020), No. 9, 2295–2333

DOI: 10.2140/ant.2020.14.2295

Abstract

We compute the Brauer group of $ℳ_{1, 1}$ , 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 $S_{3}$ 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

Vol. 14, No. 9, 2020