Stationary phase analysis for analytic newvectors and application to subconvexity problems

In this paper, we extend the results of Michel-Venkatesh and Hu-Michel-Nelson to establish an upper bound for triple product and Rankin-Selberg L-functions of the form $$L(\pi_1 \otimes \pi_2 \otimes \pi_3,\frac{1}{2})\ll_{\pi_3,\epsilon}C(\pi_1\otimes\pi_2)^{\frac{1}{2} + \epsilon} \left( \frac{C(\pi_1 \otimes \pi_2)}{C(\pi_2 \otimes \pi_2)}\right)^{-\delta}$$ in the spectral aspect, allowing conductor dropping. In particular, we obtain a subconvexity bound when $\pi_1\otimes\pi_2$ stays uniformly away from QUE-like case. The new ingredient is a stationary phase analysis of the analytic newvectors introduced by Jana and Nelson in \cite{JN19}, for both $\mathrm{PGL}_2(\mathbb{R})$ and $\mathrm{PGL}_2(\mathbb{C})$, which is applied to a test vector conjecture for local triple product periods.