Hybrid subconvexity bound for \(\mathrm{GL}(3) \times \mathrm{GL}(2)\) \(L\)-functions: \(t\) and level aspect (Q6624874)

Let \( \pi \) be a Hecke-Maass cusp form for \( \mathrm{SL}(3,\mathbb{Z}) \) and let \( f \) be a holomorphic cusp form for the congruence subgroup \( \Gamma_0(p_1p_2) \) where \( p_1<p_2 \) are distinct primes. In this article, the authors study the central values of the Rankin-Selberg \( L \)-functions \( L(1/2+it,\pi\times f)\) associated with the forms \( \pi \) and \( f \) using the delta symbol approach in the hybrid \( \mathrm{GL}(2) \)-level and \( t \)-aspects. They obtain the nontrivial bound\N\[\NL(1/2+it,\pi\times f) \ll_{\pi,\varepsilon} Q^{1/4}\left(\frac{p_1}{p_2t^2}\right)^{3/40+\varepsilon}\N\]\NHere, \( Q=(p_1p_2)^3\left(1+\left|t\right|\right)^6 \) denotes the analytic conductor of the \( L \)-function \(L(1/2+it,\pi\times f). \) This bound is ``subconvex'' in the range \( p_1<p_2<p_1t^2. \)\N\NThey apply the technique of separation of oscillation coupled with ``arithmetic'' and ``analytic'' conductor lowering trick laid out by \textit{R. Munshi} in [Am. J. Math. 137, No. 3, 791--812 (2015; Zbl 1344.11042); J. Am. Math. Soc. 28, No. 4, 913--938 (2015; Zbl 1354.11036)] to prove their main theorem quoted above, and analysis of integral transforms forms the technical heart of this paper.