On Fourier coefficients of Maass cusp forms in 3-dimensional hyperbolic space

1994 ◽  
Vol 104 (1) ◽  
pp. 77-92 ◽  
Author(s):  
S. Raghavan ◽  
J. Sengupta
Keyword(s):  
Hyperbolic Space ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
3 Dimensional

Spectral theory on 3-dimensional hyperbolic space and Hermitian modular forms

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
Roland Matthes ◽  
Yoshinori Mizuno
Keyword(s):  
Explicit Formula ◽  
Eisenstein Series ◽  
Spectral Theory ◽  
Hyperbolic Space ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Converse Theorem ◽  
Analytic Proof ◽  
3 Dimensional ◽  
Hermitian Modular Forms

AbstractWe study some arithmetics of Hermitian modular forms of degree two by applying the spectral theory on 3-dimensional hyperbolic space. This paper presents three main results: (1) a 3-dimensional analogue of Katok–Sarnak's correspondence, (2) an analytic proof of a Hermitian analogue of the Saito–Kurokawa lift by means of a converse theorem, (3) an explicit formula for the Fourier coefficients of a certain Hermitian Eisenstein series.

scholarly journals On triple correlations of Fourier coefficients of cusp forms

2018 ◽  
Vol 183 ◽  
pp. 485-492 ◽  
Author(s):  
Guangshi Lü ◽  
Ping Xi
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

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