scholarly journals HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

2013 ◽  
Vol 09 (03) ◽  
pp. 713-728 ◽  
Author(s):  
NICKOLAS ANDERSEN
Keyword(s):  
Partition Function ◽  
Generating Functions ◽  
Hecke Operator ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Powers Of 2 ◽  
Smallest Parts Function

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).

scholarly journals Traces of singular values of Hauptmoduln

2015 ◽  
Vol 11 (03) ◽  
pp. 1027-1048 ◽  
Author(s):  
Lea Beneish ◽  
Hannah Larson
Keyword(s):  
Modular Forms ◽  
Generating Functions ◽  
Singular Values ◽  
Important Paper ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Class Polynomials

In an important paper, Zagier proved that certain half-integral weight modular forms are generating functions for traces of polynomials in the j-function. It turns out that Zagier's work makes it possible to algorithmically compute Hilbert class polynomials using a canonical family of modular forms of weight [Formula: see text]. We generalize these results and consider Hauptmoduln for levels 1, 2, 3, 5, 7, and 13. We show that traces of singular values of polynomials in Hauptmoduln are again described by coefficients of half-integral weight modular forms. This realization makes it possible to algorithmically compute class polynomials.

Hecke operators on certain subspaces of integral weight modular forms

2014 ◽  
Vol 10 (07) ◽  
pp. 1909-1919 ◽  
Author(s):  
Matthew Boylan ◽  
Kenny Brown
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Invariant Subspaces ◽  
Hecke Operators ◽  
Dedekind Eta Function ◽  
Nagoya Math ◽  
Half Integral Weight ◽  
Fixed Weight ◽  
Integral Weight

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL 2(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2(ℤ)-forms of fixed weight.

scholarly journals On the Kohnen plus space for Hilbert modular forms of half-integral weight I

2013 ◽  
Vol 149 (12) ◽  
pp. 1963-2010 ◽  
Author(s):  
Kaoru Hiraga ◽  
Tamotsu Ikeda
Keyword(s):  
Modular Form ◽  
Fourier Coefficient ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Jacobi Forms ◽  
Hecke Operator ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular

AbstractIn this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

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