Subconvexity for $\mathrm{G}{\mathrm{L}}_2\times \mathrm{G}{\mathrm{L}}_2$ $L$-functions in the depth aspect

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $q=p^n$. For any given $\epsilon >0$, we establish the following subconvexity bound \begin{equation*} L(1/2,f\otimes g \otimes χ)\ll_{f,g,\varepsilon}q^{9/10+\varepsilon}. \end{equation*} The proof employs the DFI circle method with standard manipulations, including the conductor-lowering mechanism, Voronoi summation, and Cauchy–Schwarz inequality. The key input is certain estimates on the resulting character sums, obtained using the $p$-adic version of the van der Corput method.