scholarly journals The arithmetic of vector-valued modular forms on Γ0(2)

2019 ◽  
Vol 16 (02) ◽  
pp. 241-289
Author(s):  
Richard Gottesman
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Fourier Coefficients ◽  
Weight Vector ◽  
Free Module ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
Minimal Weight ◽  
Gaussian Hypergeometric Series ◽  
Vector Valued

Let [Formula: see text] denote an irreducible two-dimensional representation of [Formula: see text] The collection of vector-valued modular forms for [Formula: see text], which we denote by [Formula: see text], form a graded and free module of rank two over the ring of modular forms on [Formula: see text], which we denote by [Formula: see text] For a certain class of [Formula: see text], we prove that if [Formula: see text] is any vector-valued modular form for [Formula: see text] whose component functions have algebraic Fourier coefficients then the sequence of the denominators of the Fourier coefficients of both component functions of [Formula: see text] is unbounded. Our methods involve computing an explicit basis for [Formula: see text] as a [Formula: see text]-module. We give formulas for the component functions of a minimal weight vector-valued form for [Formula: see text] in terms of the Gaussian hypergeometric series [Formula: see text], a Hauptmodul of [Formula: see text], and the Dedekind [Formula: see text]-function.

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.

A magnetic modular form

2019 ◽  
Vol 15 (05) ◽  
pp. 907-924
Author(s):  
Yingkun Li ◽  
Michael Neururer
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Shimura Lift ◽  
Divisibility Property ◽  
Vector Valued

In this paper, we prove a conjecture of Broadhurst and Zudilin concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds and Hecke operators on vector-valued modular forms developed by Bruinier and Stein. Furthermore, we construct a family of meromorphic modular forms with this property.

scholarly journals On vector-valued modular forms and their Fourier coefficients

2003 ◽  
Vol 110 (2) ◽  
pp. 117-124 ◽  
Author(s):  
Marvin Knopp ◽  
Geoffrey Mason
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Vector Valued

scholarly journals Equivariant functions and vector-valued modular forms

2014 ◽  
Vol 10 (04) ◽  
pp. 949-954 ◽  
Author(s):  
Hicham Saber ◽  
Abdellah Sebbar
Keyword(s):  
Differential Equations ◽  
Modular Forms ◽  
Discrete Group ◽  
Complex Representation ◽  
Two Dimensional ◽  
Equivariant Functions ◽  
Vector Valued

For any discrete group Γ and any two-dimensional complex representation ρ of Γ, we introduce the notion of ρ-equivariant functions, and we show that they are parametrized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.

scholarly journals Vector-valued modular forms and the modular orbifold of elliptic curves

2016 ◽  
Vol 13 (01) ◽  
pp. 39-63 ◽  
Author(s):  
Luca Candelori ◽  
Cameron Franc
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Vector Bundles ◽  
Projective Line ◽  
Free Module ◽  
Weighted Projective Line ◽  
Holomorphic Vector Bundles ◽  
Vector Valued ◽  
Standard Techniques

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

ON THE POLYNOMIAL xd + ax + b OVER 𝔽q AND GAUSSIAN HYPERGEOMETRIC SERIES

2013 ◽  
Vol 09 (07) ◽  
pp. 1753-1763 ◽  
Author(s):  
RUPAM BARMAN ◽  
GAUTAM KALITA
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
American Mathematical Society ◽  
Polynomial Equation ◽  
Mathematical Society ◽  
Number Of Solutions ◽  
Special Values ◽  
Regional Conference ◽  
Gaussian Hypergeometric Series ◽  
Cbms Regional Conference Series

For d ≥ 2, denote by Pd(x) the polynomial over 𝔽q given by [Formula: see text]. We explicitly find the number of solutions in 𝔽q of the polynomial equation Pd(x) = 0 in terms of special values of dFd-1 and d-1Fd-2 Gaussian hypergeometric series with characters of orders d and d - 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity : Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of n+1Fn Gaussian hypergeometric series for n > 2.

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