The Brauer group of the moduli stack of elliptic curves

TL;DR

This paper calculates the Brauer group of the moduli stack of elliptic curves over various fields and rings, using advanced cohomological methods and explicit computations related to elliptic curves and Galois extensions.

Contribution

It provides the first comprehensive computation of the Brauer group of the moduli stack of elliptic curves over multiple base schemes, employing novel cohomological techniques and explicit algebraic calculations.

Findings

Brauer group over the integers and localizations computed

Explicit description of the Brauer group over finite and algebraically closed fields

Development of methods combining spectral sequences and Galois cohomology

Abstract

We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, 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 of curves with full (naive) level $2$ structure, the study of the descent spectral sequence in \'etale 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 the field of rational numbers, the computation of Hilbert symbols in the ramified case for the primes $2$ and $3$, and finding $p$-adic elliptic curves with specified properties.