GENERALIZED MODULAR FUNCTIONS AND THEIR FOURIER COEFFICIENTS

Number Theory ◽  
2009 ◽  
Author(s):  
WINFRIED KOHNBN
Keyword(s):  
Fourier Coefficients ◽  
Modular Functions

Fourier coefficients of cusp forms

1986 ◽  
Vol 100 (1) ◽  
pp. 5-29 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Field Of Study ◽  
Important Paper ◽  
Present Survey ◽  
Arithmetical Functions ◽  
Survey Article ◽  
Order Of Magnitude

The object of this survey article is to trace the influence on the theory of modular forms of the ideas contained in L. J. Mordell's important paper ‘On Mr Ramanujan's empirical expansions of modular functions’, which appeared in October 1917 in this Society's Proceedings [32]. The equally important paper [42] by S. Ramanujan, ‘On certain arithmetical functions’, referred to in Mordell's title, was published in May 1916 in the same Society's older journal, the Transactions, which was regrettably suppressed in 1928, 107 years after its foundation. Ramanujan's paper was concerned not only with multiplicative properties of Fourier coefficients of modular forms, but also with their order of magnitude. Since subsequent papers on the latter subject have also appeared in the Proceedings, it seems appropriate to include further developments in this field of study in the present survey.

Some remarks on the Fourier coefficients of cusp forms

2020 ◽  
Vol 16 (09) ◽  
pp. 1935-1943
Author(s):  
Balesh Kumar ◽  
Jay Mehta ◽  
G. K. Viswanadham
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Modular Functions ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

In this paper, we consider the angular changes of Fourier coefficients of half integral weight cusp forms and sign changes of [Formula: see text]-exponents of generalized modular functions.

ARITHMETIC PROPERTIES OF TRACES OF SINGULAR MODULI ON CONGRUENCE SUBGROUPS

2010 ◽  
Vol 06 (08) ◽  
pp. 1755-1768 ◽  
Author(s):  
SOON-YI KANG ◽  
CHANG HEON KIM
Keyword(s):  
Modular Form ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Modular Functions ◽  
Congruence Subgroups ◽  
Arithmetic Properties ◽  
Holomorphic Modular Form ◽  
Singular Moduli ◽  
Arbitrary Level

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

2016 ◽  
Vol 152 (6) ◽  
pp. 1159-1197
Author(s):  
Yingkun Li
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Maass Forms ◽  
Stark's Conjecture ◽  
Hilbert Modular ◽  
Harmonic Maass Forms ◽  
The Individual

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

scholarly journals ON FOURIER COEFFICIENTS OF MODULAR FORMS

2010 ◽  
Vol 83 (1) ◽  
pp. 50-62
Author(s):  
C. J. CUMMINS ◽  
N. S. HAGHIGHI
Keyword(s):  
Number Theory ◽  
Initial Data ◽  
Modular Forms ◽  
Recurrence Relations ◽  
Fourier Coefficients ◽  
Private Communication ◽  
Modular Functions ◽  
Genus Zero ◽  
Zero Group ◽  
Compositio Math

AbstractRecursive formulae satisfied by the Fourier coefficients of meromorphic modular forms on groups of genus zero have been investigated by several authors. Bruinier et al. [‘The arithmetic of the values of modular functions and the divisors of modular forms’, Compositio Math. 140(3) (2004), 552–566] found recurrences for SL(2,ℤ); Ahlgren [‘The theta-operator and the divisors of modular forms on genus zero subgroups’, Math. Res. Lett.10(5–6) (2003), 787–798] investigated the groups Γ0(p); Atkinson [‘Divisors of modular forms on Γ0(4)’, J. Number Theory112(1) (2005), 189–204] considered Γ0(4), and S. Y. Choi [‘The values of modular functions and modular forms’, Canad. Math. Bull.49(4) (2006), 526–535] found the corresponding formulae for the groups Γ+0(p). In this paper we generalize these results and find recursive formulae for the Fourier coefficients of any meromorphic modular form f on any genus-zero group Γ commensurable with SL(2,ℤ) , including noncongruence groups and expansions at irregular cusps. The form of the recurrence relations is well suited for the computation of the Fourier coefficients of the functions and forms on the groups which occur in monstrous and generalized moonshine. The required initial data has, in many cases, been computed by Norton (private communication).

Image Reconstruction with the Fourier Coefficients for Magnetic Induction Tomography

2020 ◽  
Vol 16 (2) ◽  
pp. 156-163
Author(s):  
Jingwen Wang ◽  
Xu Wang ◽  
Dan Yang ◽  
Kaiyang Wang
Keyword(s):  
Inverse Problem ◽  
Image Reconstruction ◽  
Magnetic Induction ◽  
Fourier Coefficients ◽  
Signal Reconstruction ◽  
Reconstruction Algorithm ◽  
Reconstruction Method ◽  
Measurement Equation ◽  
Image Reconstruction Method ◽  
Magnetic Induction Tomography

Background: Image reconstruction of magnetic induction tomography (MIT) is a typical ill-posed inverse problem, which means that the measurements are always far from enough. Thus, MIT image reconstruction results using conventional algorithms such as linear back projection and Landweber often suffer from limitations such as low resolution and blurred edges. Methods: In this paper, based on the recent finite rate of innovation (FRI) framework, a novel image reconstruction method with MIT system is presented. Results: This is achieved through modeling and sampling the MIT signals in FRI framework, resulting in a few new measurements, namely, fourier coefficients. Because each new measurement contains all the pixel position and conductivity information of the dense phase medium, the illposed inverse problem can be improved, by rebuilding the MIT measurement equation with the measurement voltage and the new measurements. Finally, a sparsity-based signal reconstruction algorithm is presented to reconstruct the original MIT image signal, by solving this new measurement equation. Conclusion: Experiments show that the proposed method has better indicators such as image error and correlation coefficient. Therefore, it is a kind of MIT image reconstruction method with high accuracy.

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