Weyl Subconvexity for $\mathrm{GL}_2$ with Simple Supercuspidal Ramification

We establish Weyl-type subconvexity bounds in the level aspect for cuspidal automorphic representations of $\mathrm{GL}_2/F$ with simple supercuspidal ramification at a prime ideal $\mathfrak{q}$. More precisely, for the family $\mathcal{F}_t^\zeta(\mathfrak{q}^3;\omega)$ consisting of representations of conductor $\mathfrak{q}^3$, central character $\omega$, and prescribed simple supercuspidal local component, we prove the fourth moment estimate \begin{align*} \sum_{\substack{\pi \in \mathcal{F}_{t}^{\zeta}(\mathfrak{q}^3;\omega) \\ C_v(\pi) \leq \mathbf{C}_v,\ v \mid \infty}} |L(1/2,\pi)|^4 \ll_{F,\varepsilon} \mathbf C_\infty^{1+\varepsilon} N_F(\mathfrak{q})^{2+\varepsilon}. \end{align*} As a consequence, we deduce the Weyl-type bound \begin{align*} L(1/2,\pi) \ll_{F,\varepsilon} C_{\infty}(\pi)^{1/4+\varepsilon} C_{\mathrm{fin}}(\pi)^{1/6+\varepsilon}. \end{align*} In particular, this bound applies to a genuinely non-self-dual family of odd conductor exponent, beyond the reach of the cubic moment method.