The Brauer group of the moduli stack of elliptic curves

Abstract

We compute the Brauer group of ${\mathcal{ℳ}}_{1,1}$, the moduli stack of elliptic curves, over $Spec\mathbb{Z}$, 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 $\mathcal{ℳ}\left(2\right)$ 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 $\mathbb{Q}$, the computation of Hilbert symbols in the ramified case for the primes $2$ and $3$, and finding $p$-adic elliptic curves with specified properties.