Subconvexity for symmetric square $L$-functions

Abstract

Let f be a holomorphic modular form of prime level p and trivial nebentypus. We show that there exists a computable δ > 0, such that L 1 2 , Sym 2 f ≪ p 1 2 −δ , with the implied constant depending only on δ and the weight of f .

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

“…In the diagonal-terms r 1 t 2 = r 2 t 1 , we save |L|C 2 which is greater than C 2 . For the off diagonal terms we note that the shift pq(r 1 t 2 − r 2 t 1 ) (defined in (6.3)) is a multiple of p. Munshi [29] encounters a similar problem of bounding…”

“…We use the version due to Heath-Brown in [10]. The following Lemma is from Munshi's paper [29,Lemma 23]. In this section we shall use the letter q to denote the moduli in the circle method.…”

“…In this case, the crux of the proof is the solution to a shifted convolution problem (6.3), where the shifts are multiples of levels of the modular form. Munshi encounters a very similar problem in his work on the symmetric square L-function [29]. We state this below in the form of a theorem, as this might be of independent interest in connection to other problems.…”

“…In a similar vein Aggarwal, Holowinsky, Lin and Sun simplified Munshi's work in [1]. Munshi has also used the circle method to obtain sub-convexity bounds for the symmetric square L-function of holomorphic GL(2) cusp forms, in the level aspect [29].…”

Subconvexity for twisted 𝐿-functions on 𝐺𝐿₃ over the Gaussian number field

“…Another approach to the GL 3 subconvexity problem is the circle method technique elaborated in the series 'The circle method and bounds for L-functions I-IV' of Munshi [Mun1]- [Mun4]. His method was further developed and simplified in [Mun5], [Mun6], [HN], [Lin] and [SZ]. Moreover, as an application of the GL 3 Kuznetsov formula in [But], Blomer and Buttcane [BB2,BB3] successfully solved the subconvexity problem in the aspect of the GL 3 Archimedean Langlands parameter (the µ-aspect).…”