scholarly journals Modular iterated integrals associated with cusp forms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Diamantis
Keyword(s):  
Modular Forms ◽  
Higher Order ◽  
New Method ◽  
Independent Interest ◽  
Cusp Forms ◽  
Fuchsian Groups ◽  
Modular Invariant ◽  
Iterated Integrals ◽  
Invariant Functions

Abstract We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.

scholarly journals A sequence of modular forms associated with higher-order derivatives of Weierstrass-type functions

Filomat ◽  
2016 ◽  
Vol 30 (12) ◽  
pp. 3253-3263
Author(s):  
Ahmet Aygunes ◽  
Yılmaz Simsek ◽  
H.M. Srivastava
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Higher Order ◽  
Cusp Forms ◽  
Derivatives Of

In this article, we first determine a sequence {fn(?)}n?N of modular forms with weight 2nk+4(2n-1-1) (n?N; k?N\{1}; N := {1,2,3,...}). We then present some applications of this sequence which are related to the Eisenstein series and the cusp forms. We also prove that higher-order derivatives of the Weierstrass type }2n-functions are related to the above-mentioned sequence {fn(?)}n?N of modular forms.

scholarly journals Three-loop contributions to the ρ parameter and iterated integrals of modular forms

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Samuel Abreu ◽  
Matteo Becchetti ◽  
Claude Duhr ◽  
Robin Marzucca
Keyword(s):  
Modular Forms ◽  
Iterated Integrals

scholarly journals Modular invariant models of leptons at level 7

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Gui-Jun Ding ◽  
Stephen F. King ◽  
Cai-Chang Li ◽  
Ye-Ling Zhou
Keyword(s):  
Modular Forms ◽  
Galois Field ◽  
Special Linear Group ◽  
Type I ◽  
Time Level ◽  
Projective Special Linear Group ◽  
Modular Invariant ◽  
Seesaw Mechanism ◽  
Linearly Independent ◽  
First Time

Abstract We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

Cuspidality in higher genus

2016 ◽  
Vol 12 (08) ◽  
pp. 2043-2060
Author(s):  
Dania Zantout
Keyword(s):  
Linear Operator ◽  
Half Space ◽  
Modular Forms ◽  
Cusp Forms ◽  
General Notion ◽  
Global Operator ◽  
Higher Genus ◽  
Genus Formula ◽  
Holomorphic Modular Forms

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.

ON THE FOURIER COEFFICIENTS OF SIEGEL MODULAR FORMS

2016 ◽  
Vol 234 ◽  
pp. 1-16
Author(s):  
SIEGFRIED BÖCHERER ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Main Tool ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Growth Properties

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

Computations with modular forms and Galois representations

2011 ◽  
Author(s):  
Johan Bosman
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Cusp Forms ◽  
Space Of Cusp Forms

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work towards the computation of polynomials that give the Galois representations associated with modular forms. Throughout, the chapter denotes the space of cusp forms of weight k, group Γ‎₁(N), and character ε‎ by Sₖ(N, ε‎).

A New Method for High-Degree Spline Interpolation: Proof of Continuity for Piecewise Polynomials

2019 ◽  
Vol 63 (3) ◽  
pp. 655-669
Author(s):  
A. Pepin ◽  
S. S. Beauchemin ◽  
S. Léger ◽  
N. Beaudoin
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Spline Interpolation ◽  
Higher Order ◽  
New Method ◽  
System Of Equations ◽  
Piecewise Polynomials ◽  
Odd Degree ◽  
High Degree

AbstractEffective and accurate high-degree spline interpolation is still a challenging task in today’s applications. Higher degree spline interpolation is not so commonly used, because it requires the knowledge of higher order derivatives at the nodes of a function on a given mesh.In this article, our goal is to demonstrate the continuity of the piecewise polynomials and their derivatives at the connecting points, obtained with a method initially developed by Beaudoin (1998, 2003) and Beauchemin (2003). This new method, involving the discrete Fourier transform (DFT/FFT), leads to higher degree spline interpolation for equally spaced data on an interval $[0,T]$. To do this, we analyze the singularities that may occur when solving the system of equations that enables the construction of splines of any degree. We also note an important difference between the odd-degree splines and even-degree splines. These results prove that Beaudoin and Beauchemin’s method leads to spline interpolation of any degree and that this new method could eventually be used to improve the accuracy of spline interpolation in traditional problems.

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