The Brauer Group of ${\mathcal{Y}}_0(2)$

We determine the Brauer group of the Deligne-Mumford stack ${\mathcal{Y}}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for ${\mathcal{M}}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of ${\mathcal{M}}_{1,1}$. We combine techniques from both papers to determine the Brauer group of ${\mathcal{Y}}_0(2)$.