Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect

TL;DR

This paper proves a subconvexity bound for certain $L$-functions associated with $ m GL_2 imes GL_2$ in the depth aspect, using advanced analytic techniques to improve bounds in number theory.

Contribution

It introduces a novel application of the $p$-adic van der Corput method combined with the circle method to achieve the subconvexity bound.

Findings

Established the bound $L(1/2,f imes g imes ext{chi}) \

Used the $p$-adic van der Corput method for character sum estimates

Demonstrated effectiveness of the conductor-lowering and Voronoi summation techniques

Abstract

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $q=p^n$. For any given $\varepsilon>0$, we establish the following subconvexity bound \begin{equation*} L(1/2,f\otimes g \otimes \chi)\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.