Level Reciprocity in the Twisted Second Moment of Rankin–selberg ‐functions

Abstract

We prove an exact formula for the second moment of Rankin-Selberg L-functions L( 1 2 , f × g) twisted by λ f (p), where g is a fixed holomorphic cusp form and f is summed over automorphic forms of a given level q. The formula is a reciprocity relation that exchanges the twist parameter p and the level q. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.

Periods and Reciprocity I

“…Let σ cuspidal satisfying Hypothesis 1 and 2, q a prime ideal of norm q and let ε > 0. There exists a positive constant λ = λ(σ, F, ε) such that for q large enough in terms of σ and ε, we have π cuspidal c(π) = q : L Sym 2 (σ) ⊗ π, 1 2 = 0 L(π, 1 2 ) = 0 λq 1−2ϑ 6 −ε .…”

“…Let p, q two prime ideals of O F of norm p and q respectively and π 1 , π 2 be two unitary automorphic representations of PGL 2 (A F ) which are unramified at all finite places and with π 1 cuspidal. In our previous paper [30], we proved, using a method related to adelic periods of automorphic forms, that the work of Andersen and Kiral [1] holds more generally over any number fields and for the first moment of a triple product L-function. We obtained a relation of the shape M(π 1 , π 2 , q, p) := π cond(π)|q L(π ⊗ π 1 ⊗ π 2 , 1 2 )λ π (p)ε π (q) q p 1/2 π cond(π)|p L(π ⊗ π 1 ⊗ π 2 , 1 2 )λ π (q)ε π (p), (1.1) where λ π (p) is the eigenvalue of the Hecke operator T p and ε π (q) ∈ {−1, +1} denotes the Atkin-Lehner eigenvalue at q.…”

“…In our previous paper [30], we proved, using a method related to adelic periods of automorphic forms, that the work of Andersen and Kiral [1] holds more generally over any number fields and for the first moment of a triple product L-function. We obtained a relation of the shape M(π 1 , π 2 , q, p) := π cond(π)|q L(π ⊗ π 1 ⊗ π 2 , 1 2 )λ π (p)ε π (q) q p 1/2 π cond(π)|p L(π ⊗ π 1 ⊗ π 2 , 1 2 )λ π (q)ε π (p), (1.1) where λ π (p) is the eigenvalue of the Hecke operator T p and ε π (q) ∈ {−1, +1} denotes the Atkin-Lehner eigenvalue at q. We recover the result of Andersen and Kiral in the particular case where F = Q and π 2 = 1 ⊞ 1 is the Eisenstein series induced from two trivial characters.…”

Mixed moment of GL(2) and GL(3)L‐functions

“…This result was refined by Young [31] and Bettin [5] and extended to the case of rational function fields by Djankovic [18]. Much activity of this type is currently being pursued for different families of L-functions, including Rankin-Selberg L-functions [1], cusp form L-functions [9,10] and triple product L-functions over a number field [33].…”

On the Random Wave Conjecture for Eisenstein Series

“…In the context of the fourth moment problem, this trick with the root number was used in a slightly different way but to the same effect in [9] and [8]. The trick has been known for some time (see [26]), and has also appeared more recently in [6,1,4].…”