Hybrid bounds for Rankin–Selberg L-functions

Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions

“…The works involving estimates of these sums with the divisor functions, known as the additive divisor problems, have a long history (See [DFIw94,Mo94] and the reference given there). Shifted convolution sums with two Fourier coefficients have been extensively investigated in [BlHa14,DFIw93,Ha03,HaMi06,Mi04] for the case when λ g (n) = λ f (n) or λ g (n) = λ f (n) and [HoMu13,HoZh17] for the general case. Thus it is natural to compare the strategies of the existing approaches and ponder about the tools necessary to study the hybrid level aspect problem.…”

“…The hybrid range 0 < η < 1 of Blomer and Harcos is stronger than our range 0 < η < 2/5. However we emphasize that our technique does not require amplification or the large sieve inequality and recently this method is adopted in [HoZh17] to extend a hybrid subconvexity range bound for…”

Sub-convexity problem for Rankin-Selberg $L$-functions

“…Holowinsky and Munshi [11] obtained a sub-convexity bound for this problem as long as p ≤ q η , with η = 2 21 . Hou and Zhang extended this to η = 2 15 [14]. Assuming that the form with the smaller level is holomorphic Zhilin Ye [33] proves a sub-convexity bound for all η.…”

Hybrid Level Aspect Subconvexity for $GL(2)\times GL(1)$ Rankin-Selberg $L$-Functions

“…The hybrid range 0 < η < 1 of Blomer and Harcos is stronger than our range 0 < η < 2/5. However we emphasize that our technique does not require amplification or Large Sieve inequality and recently this method is adopted in [11] to extend a hybrid subconvexity range bound for L(1/2, g ⊗ h) where g is a primitive holomorphic cusp form of level M and h is a primitive either holomorphic or Mass cusp form of level P with (M, P ) = 1, M a squarefree integer, and P a prime. Of course, one has the ability to push the analysis further, in either method, by analyzing the resulting sum of Kloosterman sums through Large Sieve inequality similar to the work of Z. Ye.…”