scholarly journals Arthur Parameters and Fourier coefficients for Automorphic Forms on Symplectic Groups

2016 ◽  
Vol 66 (2) ◽  
pp. 477-519 ◽  
Author(s):  
Dihua Jiang ◽  
Baiying Liu
Keyword(s):  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Symplectic Groups

On Fourier coefficients of automorphic forms of symplectic groups

2003 ◽  
Vol 111 (1) ◽  
pp. 1-16 ◽  
Author(s):  
D. Ginzburg ◽  
S. Rallis ◽  
D. Soudry
Keyword(s):  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Symplectic Groups

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.

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