scholarly journals Some Density Results on Sets of Primes for Hecke Eigenvalues

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Aiyue Zou ◽  
Huixue Lao ◽  
Shu Luo
Keyword(s):  
Positive Integer ◽  
Cusp Forms ◽  
Linear Combinations ◽  
Hecke Eigenvalues ◽  
Holomorphic Cusp Forms ◽  
Density Results

Let f and g be two distinct holomorphic cusp forms for S L 2 ℤ , and we write λ f n and λ g n for their corresponding Hecke eigenvalues. Firstly, we study the behavior of the signs of the sequences λ f p λ f p j for any even positive integer j . Moreover, we obtain the analytic density for the set of primes where the product λ f p i λ f p j is strictly less than λ g p i λ g p j . Finally, we investigate the distribution of linear combinations of λ f p j and λ g p j in a given interval. These results generalize previous ones.

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 On certain abelian varieties obtained from new forms of weight 2 on Γ0(34) and Γ0(35)

1976 ◽  
Vol 62 ◽  
pp. 29-39 ◽  
Author(s):  
Masao Koike
Keyword(s):  
Positive Integer ◽  
Abelian Varieties ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Let N be a positive integer and let Γ0(N) be the subgroup of SL(2, Z) defined by all matrices with c ≡ 0 (mod N). Let S2(Γ0(N)) be the space of holomorphic cusp forms of weight 2 with respect to Γ0(N) and let be the “essential part” of S2(Γ0(N)), which is defined in [1].

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.

scholarly journals -ADIC EISENSTEIN SERIES AND -FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

2014 ◽  
Vol 15 (3) ◽  
pp. 471-510 ◽  
Author(s):  
Ellen Eischen ◽  
Xin Wan
Keyword(s):  
Positive Integer ◽  
Eisenstein Series ◽  
Constant Term ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Imaginary Field

We construct$p$-adic families of Klingen–Eisenstein series and$L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime$p$on definite unitary groups of signature$(r,0)$(for any positive integer$r$) for a quadratic imaginary field${\mathcal{K}}$split at$p$. When$r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain$p$-adic$L$-function.

scholarly journals Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

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