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

Abstract

Let f f be a fixed self-contragradient Hecke–Maass form for S L ( 3 , Z ) SL(3,\mathbb Z) , and let u u be an even Hecke–Maass form for S L ( 2 , Z ) SL(2,\mathbb Z) with Laplace eigenvalue 1 / 4 + k 2 1/4+k^2 , k ≥ 0 k\geq 0 . A subconvexity bound O ( ( 1 + k ) 4 / 3 + ε ) O\big ((1+k)^{4/3+\varepsilon }\big ) in the eigenvalue aspect is proved for the central value at s = 1 / 2 s=1/2 of the Rankin–Selberg L L -function L ( s , f × u ) L(s,f\times u) . Meanwhile, a subconvexity bound O ( ( 1 + | t | ) 2 / 3 + ε ) O\big ((1+|t|)^{2/3+\varepsilon }\big ) in the t t aspect is proved for L ( 1 / 2 + i t , f ) 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 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 ≥ 1 n\geq 1 . These asymptotic expansions sharpened the classical results for n = 1 n=1 by Huxley.