scholarly journals ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

2021 ◽  
pp. 1-51
Author(s):  
Jesse Jääsaari ◽  
Stephen Lester ◽  
Abhishek Saha
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Fourier Coefficients ◽  
Equivalence Classes ◽  
Cusp Forms ◽  
Sign Changes ◽  
Fundamental Discriminant ◽  
Free Level ◽  
Siegel Cusp Form ◽  
Fundamental Matrices

Abstract Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

scholarly journals Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

2017 ◽  
Vol 13 (10) ◽  
pp. 2597-2625 ◽  
Author(s):  
S. Gun ◽  
J. Sengupta
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Sign Changes ◽  
Sign Change ◽  
Integral Weight ◽  
Zero Degree ◽  
Siegel Cusp Form

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

scholarly journals Fourier coefficients of Siegel cusp forms of degree two

1984 ◽  
Vol 93 ◽  
pp. 149-171 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

Our purpose is to prove the followingTheorem. Let k be an even integer ≥ 6. Letbe a Siegel cusp form of degree two, weight k. Then we have

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

scholarly journals Estimates for fourier coefficients of siegel cusp forms of degree two, II

1992 ◽  
Vol 128 ◽  
pp. 171-176 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Positive Definite ◽  
Cusp Forms ◽  
Integral Weight ◽  
Siegel Cusp Form

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that(1)(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain(2)

scholarly journals On LevelpSiegel Cusp Forms of Degree Two

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Hirotaka Kodama ◽  
Shoyu Nagaoka ◽  
Yoshitsugu Nakamura
Keyword(s):  
Cusp Form ◽  
Simple Formula ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Siegel Cusp Form

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with levelp.

scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

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.

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.

Cancellation of Cusp Forms Coefficients over Beatty Sequences on GL(m)

2011 ◽  
Vol 54 (4) ◽  
pp. 757-762
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Maass Cusp Form ◽  
Beatty Sequences

AbstractLet A(n1, n2, … , nm–1) be the normalized Fourier coefficients of a Maass cusp form on GL(m). In this paper, we study the cancellation of A(n1, n2, … , nm–1) over Beatty sequences.

scholarly journals On products of Fourier coefficients of cusp forms

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Eric Hofmann ◽  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Cusp Forms

AbstractThe purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form,

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