Improved subconvexity bounds for $GL(2)\times GL(3)$ and $GL(3)$ $L$-functions by weighted stationary phase

Abstract:

Let $f$ be a fixed self-contragradient Hecke–Maass form for $SL(3,\mathbb Z)$, and let $u$ be an even Hecke–Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k\geq 0$. A subconvexity bound $O\big ((1+k)^{4/3+\varepsilon }\big )$ in the eigenvalue aspect is proved for the central value at $s=1/2$ of the Rankin–Selberg $L$-function $L(s,f\times u)$. Meanwhile, a subconvexity bound $O\big ((1+|t|)^{2/3+\varepsilon }\big )$ in the $t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main techniques in the proofs, other than those used by Li, are $n$th-order asymptotic expansions of exponential integrals in the cases of the explicit first derivative test, the weighted first derivative test, and the weighted stationary phase integral, for arbitrary $n\geq 1$. These asymptotic expansions sharpened the classical results for $n=1$ by Huxley.