\(t\)-aspect subconvexity for \(\mathrm{GL} (2) \times \mathrm{GL}(2)\) \(L\)-function (Q2161336)

In the paper under review, the authors consider the problem for bounding \(\mathrm{GL}(2) \times \mathrm{GL}(2)\) \(L\)-function on the critical line in the \(t\)-aspect. Let \(f\) be a Hecke eigenform for the modular group \(\mathrm{SL}(2, \mathbb{Z})\) of weight \(k\) and let \(g\) be either a Hecke eigenform of weight \(k'\) or a weight zero Maass cusp form. The Rankin-Selberg convolution of \(f\) and \(g\) is given for \(\mathrm{Re}(s)>1\) by \[ L\left(s,f\otimes g\right)= \zeta(2s)\sum_{n=1}^\infty \lambda_f(n)\lambda_g(n) n^{-s}, \] where \(\lambda_f(n)\) and \(\lambda_g(n)\) denote the normalised \(n\)-th Fourier coefficient of \(f\) and \(g\) respectively. It is well known that the above series extends to an entire function to the whole complex plane \(\mathbb{C}\) and satisfies a functional equation relating \(s\) and \(1-s\). By the Phragmén-Lindelöf principle, we have \[ L\left(\frac{1}{2} +it, f \otimes g\right)\ll_\varepsilon (|t|+10)^{1+\varepsilon}. \] Breaking the convexity barrier in the above estimate is difficult and requires techniques of mollification which yield to the following \[ L\left(\frac{1}{2} +it, f \otimes g\right)\ll_\varepsilon (|t|+10)^{1-1/10+\varepsilon}. \] As a consequence, the authors derive the subconvexity bound for \[ L\left(\frac{1}{2} +it, f \right)\ll_\varepsilon \left(|t|+10\right)^{1/2-1/20+\varepsilon}. \]