The first moment of central values of symmetric square $L$-functions of cusp forms

2017 ◽  
Vol 177 (3) ◽  
pp. 277-291 ◽  
Author(s):  
Ming-Ho Ng
Keyword(s):  
Cusp Forms ◽  
Central Values ◽  
Symmetric Square ◽  
First Moment

On effective determination of symmetric-square lifts

2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Forms ◽  
Maass Forms ◽  
Central Values ◽  
Laplace Eigenvalue ◽  
Symmetric Square ◽  
Ramanujan Conjecture

AbstractLet F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.

Central values of GL(2) ×GL(3) Rankin–Selberg L-functions

2019 ◽  
Vol 16 (05) ◽  
pp. 941-962
Author(s):  
Qinghua Pi
Keyword(s):  
Trace Formula ◽  
Cusp Form ◽  
Cusp Forms ◽  
Central Values ◽  
Kuznetsov Trace Formula ◽  
First Moment

Let [Formula: see text] be a normalized holomorphic cusp form for [Formula: see text] of weight [Formula: see text] with [Formula: see text]. By the Kuznetsov trace formula for [Formula: see text], we obtain the twisted first moment of the central values of [Formula: see text], where [Formula: see text] varies over Hecke–Maass cusp forms for [Formula: see text]. As an application, we show that such [Formula: see text] is determined by [Formula: see text] as [Formula: see text] varies.

On effective determination of symmetric-square lifts, level aspect

2014 ◽  
Vol 11 (01) ◽  
pp. 51-65
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Forms ◽  
Maass Forms ◽  
Central Values ◽  
Laplace Eigenvalue ◽  
Holomorphic Cusp Forms ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
Level Aspect

Let F be the symmetric-square lift with Laplace eigenvalue λF(Δ) = 1 + 4μ2. Suppose that |μ| ≤ Λ. It is proved that F is uniquely determined by the central values of Rankin–Selberg L-functions L(s, F ⊗ h), where h runs over the set of holomorphic cusp forms of weight 10 and level q ≈ Λϱ+ϵ with [Formula: see text] for any ϵ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms. We also prove an unconditional result in weight aspect.

scholarly journals Symmetric square L-values and dihedral congruences for cusp forms

2010 ◽  
Vol 130 (9) ◽  
pp. 2078-2091 ◽  
Author(s):  
Neil Dummigan ◽  
Bernhard Heim
Keyword(s):  
Cusp Forms ◽  
Symmetric Square

scholarly journals Moments of the Critical Values of Families of Elliptic Curves, with Applications

2010 ◽  
Vol 62 (5) ◽  
pp. 1155-1181 ◽  
Author(s):  
Matthew P. Young
Keyword(s):  
Elliptic Curves ◽  
Large Family ◽  
Critical Values ◽  
Central Values ◽  
The Family ◽  
Order Of Magnitude ◽  
Rank 2 ◽  
Family Based ◽  
First Moment ◽  
Positive Rank

AbstractWe make conjectures on the moments of the central values of the family of all elliptic curves and on themoments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.Furthermore, as arithmetical applications, we make a conjecture on the distribution of ap's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

scholarly journals Twisted moments of GL(3) ×GL(2) L-functions

2021 ◽  
pp. 1-34
Author(s):  
Jakob Streipel
Keyword(s):  
Asymptotic Formula ◽  
Maass Forms ◽  
Central Values ◽  
Symmetric Square ◽  
Derivatives Of

We compute an asymptotic formula for the twisted moment of [Formula: see text] [Formula: see text]-functions and their derivatives. As an application, we prove that symmetric-square lifts of [Formula: see text] Maass forms are uniquely determined by the central values of the derivatives of [Formula: see text] [Formula: see text]-functions.

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