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

Abstract

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ and let $\ddot{I}‡$ be a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\hat{I}}º$ with $p$ prime and $\hat{I}º>12$. A subconvex bound for the central values of the Rankin-Selberg $L$-functions $L(s,f\otimes g\otimes \ddot{I}‡)$ is proved in the depth-aspect $$L\left(\frac{1}{2},f\otimes g\otimes \ddot{I}‡\right){\ll }_{f,g,\epsilon }{p}^{3/4}{\mathfrak{q}}^{15/16+\epsilon }.$$

15 pages. Comments are welcome!