Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup

2004 ◽  
Vol 74 (1) ◽  
pp. 253-279 ◽  
Author(s):  
H. Narita
Keyword(s):  
Lie Groups ◽  
Parabolic Subgroup ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Tube Type ◽  
Holomorphic Modular Forms ◽  
Minimal Parabolic Subgroup

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

AVERAGE SIZE OF GAPS IN THE FOURIER EXPANSION OF MODULAR FORMS

2007 ◽  
Vol 03 (02) ◽  
pp. 207-215 ◽  
Author(s):  
EMRE ALKAN
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
Gap Function ◽  
Quantitative Results ◽  
Average Size

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function.

scholarly journals Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Kathrin Bringmann ◽  
Pavel Guerzhoy ◽  
Ben Kane
Keyword(s):  
Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Holomorphic Modular Forms

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.

scholarly journals Constructions of vector-valued modular forms of rank four and level one

2020 ◽  
Vol 16 (05) ◽  
pp. 1111-1152
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Irreducible Representation ◽  
Differential Equations ◽  
Modular Forms ◽  
Modular Group ◽  
Tensor Products ◽  
Two Dimensional ◽  
Minimal Weight ◽  
Graded Module ◽  
Holomorphic Modular Forms ◽  
Vector Valued

This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the [Formula: see text]-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

scholarly journals Symmetric power functoriality for holomorphic modular forms, II

2021 ◽  
Author(s):  
James Newton ◽  
Jack A. Thorne
Keyword(s):  
Modular Forms ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Hecke Eigenform ◽  
Holomorphic Modular Forms

AbstractLet $f$ f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .

scholarly journals On the zeros of certain weakly holomorphic modular forms for Γ0+(2)

2016 ◽  
Vol 166 ◽  
pp. 298-323 ◽  
Author(s):  
SoYoung Choi ◽  
Bo-Hae Im
Keyword(s):  
Modular Forms ◽  
Weakly Holomorphic Modular Forms ◽  
Holomorphic Modular Forms

A family of weakly holomorphic modular forms for $\Gamma _0(2)$ with all zeros on a certain geodesic

2019 ◽  
Vol 190 (1) ◽  
pp. 57-74 ◽  
Author(s):  
SoYoung Choi ◽  
Bo-Hae Im
Keyword(s):  
Modular Forms ◽  
Weakly Holomorphic Modular Forms ◽  
Holomorphic Modular Forms

THE MINIMAL MODULAR FORM ON QUATERNIONIC

2020 ◽  
pp. 1-34
Author(s):  
Aaron Pollack
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Explicit Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Reductive Group ◽  
Automorphic Function ◽  
Unipotent Radical ◽  
Exceptional Groups ◽  
Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

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