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We prove that the Brauer group of the moduli stack of elliptic curves$\mathscr{M}_{1,1,k}$ over an algebraically closed field $k$ of characteristic$2$ is isomorphic to $\mathbb{Z}/(2)$. We also compute the Brauer group of$\mathscr{M}_{1,1,k}$ where $k$ is a finite field of characteristic $2$.

The Brauer group of $\mathscr{M}_{1,1}$ over algebraically closed fields of characteristic $2$