Second order average estimates on local data of cusp forms

Abstract

We specify sufficient conditions for the square modulus of the local parameters of a family of GL n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n 4. As an application, we show that Rankin-Selberg L-functions on GL n 1 × GL n 2 , for n i 4, satisfy the standard convexity bound.

Applications of the Kuznetsov formula on GL(3)

“…If ϕ is an eigenfunction of the Hecke algebra (see [ 17 , Sect. 6.4]), we define its L -function by L ( ϕ , s ):=∑ m A ϕ (1, m ) m − s , and the Rankin–Selberg L -function by It follows from [ 25 , Theorem 2] or [ 6 , Corollary 2] that the coefficients are essentially bounded on average, uniformly in ν : The space of cusp forms is equipped with an inner product It is known that can be continued holomorphically to ℂ with the exception of a simple pole at s =1 whose residue is proportional to ∥ ϕ ∥ 2 [ 17 , Theorem 7.4.9]. The proportionality constant is given in the next lemma.…”

Upper Bounds on L-Functions at the Edge of the Critical Strip

“…Molteni proved this result for all Rankin-Selberg L-functions satisfying certain strong assumptions on the size of the parameters which are currently unknown beyond GL(2). Brumley [2] extended this to m, n ≤ 4 unconditionally by replacing the condition of Molteni by the existence of a certain strong isobaric lift which is known up to GL(4). Our Theorem 2 gives that there exists some constant C > 0 such that…”

“…Finally, we mention here that this convexity bound for L(s, π 1 × π 2 ) has several interesting and immediate applications. Brumley describes an extension of the zero density result of Kowalski and Michel [14] to all cusp forms on GL(n) over Q to n ≤ 4 based on his work in [2] which we can now extend to all n. See Corollary 4 in [2] for more details. In [3], Brumley proves an effective strong multiplicity result which states that two cusp forms are the same if they agree on all the spherical non-archimedean places with norm bounded by C A for some A > 0.…”

On the global sup-norm of GL(3) cusp forms

“…Assume that φ is tempered at the archimedean place. If φ is arithmetically normalized, then (7) holds with the extra factor L(1, φ, Ad) 1/2 ≪ ε λ ε φ on the right hand side by the work of Brumley [Br1,Cor. 2] or Li [Li,Thm.…”

“…the first double sum on the right hand side of (17) can be estimated by the result of Brumley [Br1,Cor. 2] or Li [Li,Thm.…”