First moment of Rankin–Selberg central L-values and subconvexity in the level aspect

Abstract

Let 1 N < M with N and M coprime and square-free. Through classical analytic methods we estimate the first moment of central L-values L( 1 2 , f × g) where f ∈ S * k (N ) runs over primitive holomorphic forms of level N and trivial nebentypus and g is a given form of level M . As a result, we recover the bound L( 1The first moment method also applies to the special derivative L ′ ( 1 2 , f × g) under the assumption that it is non-negative for all f ∈ S * k (N ).

Rankin–Selberg L-functions and the reduction of CM elliptic curves

“…Hyrbid subconvexity bounds for more general families of Rankin-Selberg L-functions have been obtained by various methods (see e.g. [10,16,17,33]). …”

The Second Moment of Rankin-Selberg L-function and Hybrid Subconvexity Bound

“…for some absolute positive constant δ have been shown by Kowalski-Michel-VanderKam [KMV], Michel [M1], and Harcos-Michel [HM1]. Furthermore, the subconvexity bound for two independently varying forms have been established in the works of Michel-Ramakrishnan [MR], Feigon-Whitehouse [FW], Nelson [N1] and Holowinsky-Templier [HT1] in situations where positivity of the central L-values is known.…”

The Petersson/Kuznetsov trace formula with prescribed local ramifications

“…Thus it is actually possible to obtain an asymptotic formula when N 1/2 p 3l is sufficiently large compared to M. Remark 1.12. Compared to [5] [6], this result has two differences/improvements. First of all, [5] [6] assume N to be square-free.…”

“…Proof. We shall focus on the case when v p (c l ) ≤ k ≤ c (π), as the case when k ≥ c (π) will be easier (and one can use the argument in [6] with slight modifications). For conciseness we drop all ǫ-terms in our computations.…”