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

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 On the equality case of the Ramanujan Conjecture for Hilbert modular forms

2019 ◽  
Vol 15 (10) ◽  
pp. 2107-2114
Author(s):  
Liubomir Chiriac
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Complex Multiplication ◽  
Hilbert Modular Forms ◽  
Hilbert Modular Form ◽  
Automorphic Representations ◽  
Equality Case ◽  
Hilbert Modular ◽  
Ramanujan Conjecture ◽  
Satake Parameters

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

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.

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 Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

scholarly journals EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

2020 ◽  
pp. 1-36
Author(s):  
Bart Michels
Keyword(s):  
Lower Bound ◽  
Closed Geodesic ◽  
Extreme Values ◽  
Theta Series ◽  
Hyperbolic Surface ◽  
Quadratic Fields ◽  
Maass Forms ◽  
Real Quadratic Fields ◽  
Central Values ◽  
Real Quadratic

Abstract Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

scholarly journals CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

2010 ◽  
Vol 06 (02) ◽  
pp. 281-309 ◽  
Author(s):  
F. G. GARVAN
Keyword(s):  
Partition Function ◽  
Rank Function ◽  
Maass Forms ◽  
Hecke Eigenforms ◽  
Half Integer ◽  
Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

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.

scholarly journals Twisted traces of CM values of weak Maass forms

2013 ◽  
Vol 133 (6) ◽  
pp. 1827-1845 ◽  
Author(s):  
Claudia Alfes ◽  
Stephan Ehlen
Keyword(s):  
Maass Forms

scholarly journals CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS

2011 ◽  
Vol 07 (04) ◽  
pp. 855-919
Author(s):  
YUVAL Z. FLICKER
Keyword(s):  
Quadratic Extension ◽  
Summation Formula ◽  
Unipotent Radical ◽  
Automorphic Representations ◽  
Orbital Integrals ◽  
Relative Trace Formula ◽  
Almost Everywhere ◽  
Genus 2 ◽  
Technical Advances ◽  
Ramanujan Conjecture

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

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