On the universality for L-functions attached to Maass forms

Analysis ◽  
2005 ◽  
Vol 25 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Hirofumi Nagoshi
Keyword(s):  
Maass Forms ◽  
Universality Theorem ◽  
Ramanujan Conjecture ◽  
Derivatives Of

AbstractWe establish the universality theorem on some region for automorphic L-functions attached to Maass forms for SL(2, ℤ). As is well known, the Ramanujan conjecture is still open for these forms. The universality theorem for the derivatives of those L-functions and some applications are also established.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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 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.

On the Generalizations of Universality Theorem for L-Functions of Elliptic Curves

2017 ◽  
Vol 2 (44) ◽  
pp. 102-104
Author(s):  
Virginija Garbaliauskienė
Keyword(s):  
Positive Integer ◽  
Weak Convergence ◽  
Elliptic Curves ◽  
Limit Theorems ◽  
Probability Measures ◽  
Universality Theorem ◽  
Convergence Of Probability Measures ◽  
Derivatives Of

In the paper, the continuous type’s universality theorem for L-functions of elliptic curves is discussed and its generalizations in three directions – for positive integer powers and derivatives of L-functions of elliptic curves as well as the weighted universality theorem of L-functions of elliptic curves – are given. The proofs of the universality are based on limit theorems in the sense of weak convergence of probability measures in functional spaces.

Average Bound Toward the Generalized Ramanujan Conjecture and Its Applications on Sato–Tate Laws for GL(n)

2020 ◽  
Author(s):  
Yuk-Kam Lau ◽  
Ming Ho Ng ◽  
Yingnan Wang
Keyword(s):  
Maass Forms ◽  
Ramanujan Conjecture ◽  
Satake Parameters ◽  
Trivial Estimate

Abstract We give the 1st non-trivial estimate for the number of $GL(n)$ ($n\ge 3$) Hecke–Maass forms whose Satake parameters at any given prime $p$ fail the Generalized Ramanujan Conjecture and study some applications on the (vertical) Sato–Tate laws.

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.

Physical Properties of TCNQ Salts with N-Methyl Derivatives of Pyridine

1982 ◽  
Vol 85 (1) ◽  
pp. 257-263 ◽  
Author(s):  
A. Graja ◽  
M. Przybylski ◽  
B. Butka ◽  
R. Swietlik
Keyword(s):  
Physical Properties ◽  
Methyl Derivatives ◽  
Derivatives Of
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