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.

中文翻译：

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椭圆曲线模栈的布劳尔群

我们计算整数上椭圆曲线模栈的 Brauer 群、整数的局部化、奇特征的有限域和特征的代数闭域不是 $2$。所涉及的方法包括使用勒让德曲线的参数空间和全（朴素）级$2$结构的曲线模堆叠，研究etale上同调中的下降谱序列和fppf上同调中的Leray谱序列，计算$S_3$在某个积分表示中的群上同调，有理数域的三次Galois扩展的分类，质数$2$和$3$分枝情况下希尔伯特符号的计算，以及找到$p$ -adic 椭圆曲线具有指定的属性。