Subconvexity for Rankin Selberg L-Functions at Special Points

Abstract

Let $f$ and $g$ be normalized Hecke-Maass cusp forms for the full modular group having spectral parameters $t_f$ and $t_g$ respectively with $t_f,t_g\asymp T\rightarrow \infty $. In this paper we show that the Rankin Selberg $L$-function associated to the pair $(f,g)$ at the special points $t=\pm(t_f+t_g)$, satisfies the subconvex bound \begin{align*} L\left(\frac{1}{2}+it,f\otimes g\right)\ll_{\eps} T^{61/84+\eps}. \end{align*} Additionally at the points $t=\pm(t_f-t_g)\asymp T^{\nu}$ with $2/3+\eps<\nu\leq 1$ we show the subconvex bound \begin{align*} L(1/2+it,f\otimes g)\ll_\eps {T^{7/12+\nu/8+\eps}}, \; \text{if }\; 2/3+\eps< \nu\leq 14/17, \end{align*} and \begin{align*} L(1/2+it,f\otimes g)\ll_\eps {T^{1/2+19\nu/84+\eps}}, \; \text{if }\; 14/17\leq \nu\leq 1. \end{align*} With the above results we are able to address the subconvexity problem in the spectral aspect for $GL(2)\times GL(2)$ Rankin Selberg $L$-functions when the parameters of both the forms vary under the additional challenge of a considerable amount conductor dropping occurring due to the special points in question.