scholarly journals LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS

2016 ◽  
Vol 222 (1) ◽  
pp. 137-185 ◽  
Author(s):  
MASANORI MUTO ◽  
HIRO-AKI NARITA ◽  
AMEYA PITALE
Keyword(s):  
Parabolic Subgroup ◽  
Quaternion Algebra ◽  
Congruence Subgroup ◽  
Explicit Construction ◽  
Cusp Forms ◽  
Cuspidal Representation ◽  
Local Component ◽  
Ramanujan Conjecture ◽  
Cuspidal Representations

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

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.

BASES FOR Sk(Γ1(4)) AND FORMULAS FOR EVEN POWERS OF THE JACOBI THETA FUNCTION

2013 ◽  
Vol 09 (08) ◽  
pp. 1973-1993 ◽  
Author(s):  
SHINJI FUKUHARA ◽  
YIFAN YANG
Keyword(s):  
Eisenstein Series ◽  
Theta Function ◽  
Congruence Subgroup ◽  
Negative Integer ◽  
Cusp Forms ◽  
Jacobi Theta Function

We find a basis for the space Sk(Γ1(4)) of cusp forms of weight k for the congruence subgroup Γ1(4) in terms of Eisenstein series. As an application, we obtain formulas for r2k(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares.

Globalization of Distinguished Supercuspidal Representations of GL(n)

2002 ◽  
Vol 45 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Jeffrey Hakim ◽  
Fiona Murnaghan
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Cuspidal Representation ◽  
Supercuspidal Representation ◽  
Local Component ◽  
Supercuspidal Representations

AbstractAn irreducible supercuspidal representation π of G = GL(n, F), where F is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup H of G and a quasicharacter χ of H if HomH(π, χ) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

scholarly journals An explicit construction of non-tempered cusp forms on $$O(1,8n+1)$$

2019 ◽  
Vol 44 (2) ◽  
pp. 349-384
Author(s):  
Yingkun Li ◽  
Hiro-aki Narita ◽  
Ameya Pitale
Keyword(s):  
Explicit Construction ◽  
Cusp Forms

scholarly journals The modular equation and modular forms of weight one

1985 ◽  
Vol 100 ◽  
pp. 145-162 ◽  
Author(s):  
Toyokazu Hiramatsu ◽  
Yoshio Mimura
Keyword(s):  
Modular Forms ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Algebraic Integer ◽  
Complex Multiplication ◽  
Cusp Forms ◽  
Imaginary Quadratic Fields ◽  
Modular Equation ◽  
Class Invariants ◽  
Weight One

This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted byΦ(x) = 0.

scholarly journals Nonvanishing of L-functions, the Ramanujan Conjecture, and Families of Hecke Characters

2013 ◽  
Vol 65 (1) ◽  
pp. 22-51 ◽  
Author(s):  
Valentin Blomer ◽  
Farrell Brumley
Keyword(s):  
Class Group ◽  
Cusp Forms ◽  
Critical Strip ◽  
Residue Classes ◽  
Ramanujan Conjecture

AbstractWe prove a nonvanishing result for families of GLn× GLn Rankin–Selberg L-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on GLn. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

scholarly journals Period polynomials and explicit formulas for Hecke operators on Γ0(2)

2009 ◽  
Vol 146 (2) ◽  
pp. 321-350 ◽  
Author(s):  
SHINJI FUKUHARA ◽  
YIFAN YANG
Keyword(s):  
Vector Space ◽  
Explicit Form ◽  
Congruence Subgroup ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Isomorphism Theorem ◽  
Explicit Formulas ◽  
Space Of Cusp Forms

AbstractLet Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

Integral moments of automorphic L-functions

2016 ◽  
Vol 12 (07) ◽  
pp. 1827-1843
Author(s):  
Hengcai Tang ◽  
Xuanxuan Xiao
Keyword(s):  
Real Number ◽  
Riemann Hypothesis ◽  
Cuspidal Representation ◽  
Generalized Riemann Hypothesis ◽  
Number Formula ◽  
Ramanujan Conjecture

Let [Formula: see text] be a self-contragredient irreducible unitary cuspidal representation of [Formula: see text] with [Formula: see text], and [Formula: see text] be the automorphic [Formula: see text]-function attached to [Formula: see text]. Assume that [Formula: see text] is self-contragredient. Under the Generalized Ramanujan Conjecture and Generalized Riemann Hypothesis for [Formula: see text], the estimate [Formula: see text] holds for all real number [Formula: see text] as [Formula: see text].

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