Power moments of Hecke eigenvalues for congruence group

2019 ◽  
Vol 198 ◽  
pp. 139-158 ◽  
Author(s):  
Ping Song ◽  
Wenguang Zhai ◽  
Deyu Zhang
Keyword(s):  
Congruence Group ◽  
Hecke Eigenvalues ◽  
Power Moments

On Some Entire Modular Forms of Weights and for the Congruence Group Γ0(4N)

1996 ◽  
Vol 3 (5) ◽  
pp. 485-500
Author(s):  
G. Lomadze
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Congruence Group ◽  
Positive Integers

Abstract Entire modular forms of weights and for the congruence group Γ0(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.

scholarly journals The orthogonality of Hecke eigenvalues

2007 ◽  
Vol 143 (03) ◽  
pp. 541-565 ◽  
Author(s):  
Henryk Iwaniec ◽  
Xiaoqing Li
Keyword(s):  
Hecke Eigenvalues

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Target Space ◽  
Large Set ◽  
Hecke Eigenvalues ◽  
Artin's Conjecture ◽  
Number Problem ◽  
Novel Method ◽  
Hecke Modules ◽  
Primitive Roots

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (03) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Some problems involving Hecke eigenvalues

2019 ◽  
Vol 159 (1) ◽  
pp. 287-298
Author(s):  
H. F. Liu ◽  
R. Zhang
Keyword(s):  
Hecke Eigenvalues

On higher-power moments of $$\Delta_{(1)}(x)$$

2020 ◽  
Vol 162 (2) ◽  
pp. 445-464
Author(s):  
D. Liu ◽  
Y. Sui
Keyword(s):  
Higher Power ◽  
Power Moments
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