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On modular forms of half integral weight

The Annals of Mathematics, 1973
The recent development of the theory of modular forms and associated zeta functions, together with all its arithmetic significance, is quite pleasing, and our knowledge in this field is evergrowing, but the forms of half integral weight have attracted only casual attention, in spite of their importance and ancientness.
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Congruences for Γ1(4)-modular forms of half-integral weight

The Ramanujan Journal, 2006
The main result of this paper gives a complete description of the algebra of level 4, mod p modular forms of half-integral weight. We show that this is the algebra of regular functions of an affine curve over \(\overline{\mathbb F}_p\). The result parallels results of Swinnerton-Dyer (level 1), and Katz (general level) for integral weight forms.
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Coefficients of half-integral weight modular forms modulo ?j

Mathematische Annalen, 2004
For a half-integral weight cusp form \(F(z)\) with Fourier series \[ \sum_{n \geq 1} a(n)q^n, \] where the \(a(n)\) are integers, the authors say that the coefficients \(a(n)\) are \textit{well-distributed} modulo \(M\) if for every integer \(r\), \[ \#\{ 1 \leq n \leq X : a(n) \equiv r \pmod{M} \} \gg_{r,M} \begin{cases} \frac{\sqrt{X}}{\log X ...
Ahlgren, Scott, Boylan, Matthew
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Half integral weight Jacobi forms and periods¶of modular forms

manuscripta mathematica, 2001
In [Abh. Math. Semin. Univ. Hamb. 16, 1-28 (1949; Zbl 0035.06004)], \textit{G. Bol} proved: Suppose \(r\in \mathbb{Z}\), \(r\geq 0\); then \[ D^{(r+1)} \Biggl\{(c\tau+d)^r F\biggl( \frac{a\tau+b}{c\tau+d} \biggr)\Biggr\}= (c\tau+d)^{-r-2} F^{(r+1)} \biggl( \frac{a\tau+b}{c\tau+d} \biggr), \] for \(ad-bc=1\) and any \(F\) defined on the complex plane \(\
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ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT

International Journal of Number Theory, 2013
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.
Choie, YJ, Kohnen, W
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Hecke operators on modular forms of half-integral weight

Archiv der Mathematik, 1988
In the paper [Arch. Math. 32, 158-165 (1979; Zbl 0407.10022)] \textit{Hong- Jen Hsiao} and \textit{Hong-Chang Lee} have proved that the Dirichlet series associated with a modular form f of integral weight for the full modular group has an Euler product expansion if and only if f is an eigenfunction for finitely many explicitly given Hecke operators. In
Manickam, M., Ramakrishnan, B., Vasudevan, T. C.   +2 more
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Modular Forms of Half Integral Weight

2006
The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function $$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty {e^{2\pi in^2 z} } $$ and the Dedekind eta function $$ \eta \left( z \right) = e^{\pi iz/12} \prod _{n = 1}^\infty (1 - e^{2\pi inz} ) $$ are ...
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Theta function and hilbert modular forms of half integral weight

Acta Mathematica Sinica, 1995
The author constructs by means of spherical functions \(P\) with respect to a symmetric matrix \(A\) Hilbert modular forms \(f\) of half-integral weight for principal congruence subgroups of level \(2N\) of the Hilbert modular group of a totally real number field of degree \(r\) and ring of integers \({\mathcal O}\).
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A canonical subspace of modular forms of half-integral weight

Mathematische Annalen, 2009
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Gun, Sanoli, Manickam, M., Ramakrishnan, B.   +2 more
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