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.

scholarly journals Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

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].

scholarly journals On the Fourier expansions of Jacobi forms

2004 ◽  
Vol 2004 (48) ◽  
pp. 2583-2594 ◽  
Author(s):  
Howard Skogman
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Jacobi Form ◽  
The Other ◽  
Jacobi Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Fourier Expansions ◽  
The Relationship ◽  
Vector Valued

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexesp,p2, andpqfor distinct odd primesp,q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the formpqorp2, there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the formp, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices fromSl2(ℤ)to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.

ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT

2013 ◽  
Vol 09 (08) ◽  
pp. 1879-1883 ◽  
Author(s):  
YOUNG JU CHOIE ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Growth Condition ◽  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Half Integral Weight ◽  
Fundamental Discriminant ◽  
Integral Weight ◽  
Elliptic Modular Form

It is known that if the Fourier coefficients a(n)(n ≥ 1) of an elliptic modular form of even integral weight k ≥ 2 on the Hecke congruence subgroup Γ0(N)(N ∈ N) satisfy the bound a(n) ≪f nc for all n ≥ 1, where c > 0 is any number strictly less than k - 1, then f must be cuspidal. Here we investigate the case of half-integral weight modular forms. The main objective of this note is to show that to deduce that f is a cusp form, it is sufficient to impose a suitable growth condition only on the Fourier coefficients a(|D|) where D is a fundamental discriminant with (-1)kD > 0.

scholarly journals Modular descent of Siegel modular forms of half integral weight and an analogy of the Maass relation

1986 ◽  
Vol 102 ◽  
pp. 51-77 ◽  
Author(s):  
Yoshio Tanigawa
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Fourier Coefficients ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Maass Relation

In [8], H. Maass introduced the ‘Spezialschar’ which is now called the Maass space. It is defined by the relation of the Fourier coefficients of modular forms as follows. Let f be a Siegel modular form on Sp(2,Z) of weight k, and let be its Fourier expansion, where . Then f belongs to the Maass space if and only if

Sign in / Sign up

Export Citation Format

Share Document