A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

2018 ◽  
Vol 5 (3) ◽  
Author(s):  
Francis Brown
Keyword(s):  
Modular Forms ◽  
Cusp Forms ◽  
Real Analytic ◽  
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 Mock Jacobi forms in basic hypergeometric series

2009 ◽  
Vol 145 (03) ◽  
pp. 553-565 ◽  
Author(s):  
Soon-Yi Kang
Keyword(s):  
Modular Forms ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Theta Functions ◽  
Jacobi Forms ◽  
Mock Theta Functions ◽  
Torsion Points ◽  
Weakly Holomorphic Modular Forms ◽  
Real Analytic ◽  
Holomorphic Modular Forms

AbstractWe show that someq-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers ofq. We also prove that certain linear sums ofq-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

scholarly journals A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS

2020 ◽  
Vol 8 ◽  
Author(s):  
FRANCIS BROWN
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Multiple Zeta Values ◽  
New Family ◽  
Algebra Of Functions ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Upper Half Plane ◽  
Real Analytic ◽  
Holomorphic Modular Forms

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.

scholarly journals On cycle integrals of weakly holomorphic modular forms

2015 ◽  
Vol 158 (3) ◽  
pp. 439-449 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
PAVEL GUERZHOY ◽  
BEN KANE
Keyword(s):  
Modular Forms ◽  
Cusp Forms ◽  
Weakly Holomorphic Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Holomorphic Modular Forms

AbstractIn this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.

ON p-ADIC PROPERTIES OF TWISTED TRACES OF SINGULAR MODULI

2010 ◽  
Vol 06 (03) ◽  
pp. 625-653
Author(s):  
DANIEL LE ◽  
SHELLY MANBER ◽  
SHRENIK SHAH
Keyword(s):  
Modular Forms ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Cusp Forms ◽  
Imaginary Quadratic Fields ◽  
Modulo P ◽  
Holomorphic Modular Forms ◽  
Class Polynomials ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in Fp.

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.

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 .

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