scholarly journals The orthogonality of Hecke eigenvalues

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

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

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 $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

Some problems involving Hecke eigenvalues

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

scholarly journals Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

2014 ◽  
Vol 89 (4) ◽  
pp. 979-1014 ◽  
Author(s):  
Étienne Fouvry ◽  
Satadal Ganguly ◽  
Emmanuel Kowalski ◽  
Philippe Michel
Keyword(s):  
Gaussian Distribution ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Hecke Eigenvalues

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 Fourier coefficients of the overconvergent generalized eigenform associated to a CM form

2020 ◽  
Vol 16 (06) ◽  
pp. 1185-1197
Author(s):  
Chi-Yun Hsu
Keyword(s):  
Modular Form ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Hecke Eigenvalues ◽  
Hecke Eigenform ◽  
Critical Slope

Let [Formula: see text] be a modular form with complex multiplication. If [Formula: see text] has critical slope, then Coleman’s classicality theorem implies that there is a [Formula: see text]-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as [Formula: see text]. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of [Formula: see text]-adic overconvergent forms containing [Formula: see text].

scholarly journals On the Hecke eigenvalues of Maass forms

2019 ◽  
Vol 141 (2) ◽  
pp. 485-501 ◽  
Author(s):  
Wenzhi Luo ◽  
Fan Zhou
Keyword(s):  
Maass Forms ◽  
Hecke Eigenvalues

Toward an explicit eigencurve for GL(3)

2017 ◽  
Vol 13 (05) ◽  
pp. 1213-1231
Author(s):  
Avner Ash ◽  
David Pollack
Keyword(s):  
Congruence Subgroup ◽  
Weight Space ◽  
Projection Formula ◽  
Hecke Eigenvalues

Starting with a numerically noncritical (at [Formula: see text]) Hecke eigenclass [Formula: see text] in the homology of a congruence subgroup [Formula: see text] of [Formula: see text] (where [Formula: see text] divides the level of [Formula: see text]) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of [Formula: see text] to a Hecke eigenclass [Formula: see text] with coefficients in a module of [Formula: see text]-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection [Formula: see text] to weight space of the eigencurve around the point [Formula: see text] corresponding to the system of Hecke eigenvalues of [Formula: see text]. We do this under the conjecturally mild hypothesis that [Formula: see text] is smooth at [Formula: see text].

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