scholarly journals The first moment of central values of symmetric square L-functions in the weight aspect

2017 ◽  
Vol 46 (3) ◽  
pp. 775-794
Author(s):  
Shenhui Liu
Keyword(s):  
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.

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 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.

scholarly journals On the average of central values of symmetric square L-functions in weight aspect

2002 ◽  
Vol 167 ◽  
pp. 95-100 ◽  
Author(s):  
Winfried Kohnen ◽  
Jyoti Sengupta
Keyword(s):  
Modular Group ◽  
Central Values ◽  
Hecke Eigenforms ◽  
Lindelöf Hypothesis ◽  
Symmetric Square

AbstractIt is proved that the central values of symmetric square L-functions of normalized Hecke eigenforms for the full modular group on average satisfy an analogue of the Lindelöf hypothesis in weight aspect, under the assumption that these values are non-negative.

scholarly journals The second moment of the central values of the symmetric square L-functions

2014 ◽  
Vol 38 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Jonathan Wing Chung Lam
Keyword(s):  
Second Moment ◽  
Central Values ◽  
Symmetric Square

scholarly journals Bounds for twisted symmetric square L-functions via half-integral weight periods

2020 ◽  
Vol 8 ◽  
Author(s):  
Paul D. Nelson
Keyword(s):  
Modular Forms ◽  
Triple Product ◽  
Quadratic Character ◽  
Maass Form ◽  
Half Integral Weight ◽  
Fundamental Discriminant ◽  
Integral Weight ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
First Moment

Abstract We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

scholarly journals The first moment of the symmetric-square L-function

2007 ◽  
Vol 124 (2) ◽  
pp. 259-266 ◽  
Author(s):  
Rizwanur Khan
Keyword(s):  
Symmetric Square ◽  
First Moment
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