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

TL;DR

This paper establishes a subconvex bound for the central values of Rankin-Selberg L-functions associated with GL(2) forms twisted by a prime power conductor Dirichlet character, advancing understanding in the depth aspect.

Contribution

The paper provides the first subconvex bound for these L-functions in the depth aspect, specifically for prime power conductors with conductor exponent greater than 12.

Findings

Proves a subconvex bound: $L(1/2,f\otimes g\otimes \chi) \ll p^{3/4} \mathfrak{q}^{15/16+\varepsilon}$

Extends bounds to depth aspect for prime power conductors

Improves previous convexity bounds in this setting

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 $\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa>12$. A subconvex bound for the central values of the Rankin-Selberg $L$-functions $L(s,f\otimes g \otimes \chi)$ is proved in the depth-aspect $$ L\left(\frac{1}{2},f\otimes g \otimes \chi\right)\ll_{f,g,\varepsilon} p^{3/4}\mathfrak{q}^{15/16+\varepsilon}. $$