scholarly journals Fourier coefficients of automorphic forms and integrable discrete series

2016 ◽  
Vol 270 (10) ◽  
pp. 3639-3674 ◽  
Author(s):  
Goran Muić
Keyword(s):  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Discrete Series

SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2, 2) AND THEIR RESTRICTION TO Sp(1, 1)

2012 ◽  
Vol 23 (10) ◽  
pp. 1250104
Author(s):  
ATSUO YAMAUCHI ◽  
HIRO-AKI NARITA
Keyword(s):  
Hyperbolic Space ◽  
Symplectic Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Discrete Series ◽  
Theta Series ◽  
Explicit Construction ◽  
Discrete Series Representations ◽  
Vector Valued ◽  
Real Symplectic Group

In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

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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