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Questions tagged [half-integral-weight]

Questions about half-integral weight modular forms, and more generally automorphic representations associated to metaplectic groups.

3
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1answer
173 views

Why the level of a half integral weight modular form must be a multiple of 4?

Let $\Gamma_0(N)$ be the Hecke congruence subgroup. Let $S_{k+1/2}(\Gamma_0(N))$ be the space of holomorphic forms of weight $k+1/2$ on $\Gamma_0(N)$, where $k\in\mathbb{N}$. How to prove that if $S_{...
4
votes
0answers
171 views

Abscissa of conditional convergence of the $L$-series attached to a modular form of half-integral weight

Let $f$ be a cusp form of half-integral weight $k/2$ on $\Gamma_0(4N),$ let $a(n)$ its Fourier coefficient My question is: is it known the value of the abscissa of conditional convergence $\sigma_c$ ...
6
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1answer
281 views

Twisted modular forms of half-integral weight

I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of $$ \theta(\...
3
votes
0answers
227 views

Functional equation link two Dirichlet series

Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\...
3
votes
1answer
316 views

Estimate the ratio $\dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\underset{n\le X}{\sum} a(n)^2}$

Let $f=\sum_{n\ge 1} a(n)q^n\in M_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ be a modular form of half-integral wieght. Can someone prove or disprove that: $$X\ll \dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\...
0
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1answer
194 views

Clarification of the proof of the main theorem of the paper of Hulse et al

I am trying to understand some open steps in the following article The Sign of Fourier coefficients of Half-integral Weight Cusp Form by Hulse, Kiral, Kuan, and Lim, I find the following : Let $f\...
6
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1answer
402 views

A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights

Denote $\operatorname{dim}(M_k(\Gamma_0(N)))$ by $m(k,N)$ and $\operatorname{dim}(S_k(\Gamma_0(N)))$ by $s(k,N)$. Let $N$ any positive multiple of $4$ and $j \ge 1$. $$ a(N) := \frac1j \left(m \...
2
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0answers
256 views

Modular forms of half-integral weight: why do we need the cover of SL?

I wanted to read a bit about modular forms of half integral weight. There is the notion of a 'factor of automorphy' (as for example, R.Rankin uses it in the book 'Modular Forms and Functions'). Now I ...
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vote
0answers
105 views

Sign of the functional equation of L function and Shimura lift

I would like to know what happens to the root number of a half integral weight automorphic form (holomorphic or not), i.e. the sign of the functional equation of its L-function when we apply the ...
7
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1answer
507 views

Average of Fourier coefficients of a cusp form of half integral weight

Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
2
votes
0answers
189 views

Conceptual reason behind Shimura lifts

Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G =SL_2(\mathbb{R}...
5
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2answers
325 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
1
vote
1answer
267 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
8
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1answer
615 views

Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4?

Given a quadratic form $F$ in $n$ variables, there is an associated theta function $\theta_F(z) = \sum_{m \in \mathbb{Z}} q^{F(m)}$, which is a modular form of weight $n/2$. Letting $F(m) = m^2$ ...
17
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3answers
913 views

Why only half-integral weight automorphic forms?

Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or ...
11
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3answers
1k views

Do L-functions exist for Half-integral weight modular forms?

Classically, we can attach $L$-functions (with properties like, analytic continuation, functional equation) to Dirichlet characters, Hecke eigenforms, etc... My question is: can one attach $L$-...
5
votes
1answer
356 views

Half integral weight Hecke operators

I would like to find a source giving the exact formula for the product of two Hecke operators $T_{\kappa}(n^2)$ and $T_\kappa(m^2)$ of half integral weight. That is, $\kappa \in \frac 12 \mathbb{Z} - \...
12
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1answer
620 views

The space of lattices and modular forms of weight 1/2

Suppose my favorite way of thinking about modular forms is as functions on the space of (real, 2D) lattices. One can identify this space with $SL_2(\mathbb{Z}) \backslash GL_2(\mathbb{R})$, i.e. ...
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3answers
2k views

How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?

The Eisenstein series $$ G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}} $$ are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms $...
4
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1answer
559 views

Odd powers of the theta function as eigenforms

Is it "well-known" which odd powers of the theta function are eigenforms for the half-integral weight Hecke operators? If so, what is a good reference? Is there a slick algorithm for proving ...
8
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4answers
2k views

Basis for modular forms of half-integral weight

Given a character $\chi$ and $k$ odd how can one compute a basis for the space of modular forms $M_\frac{k}{2}(\Gamma_0(4),\chi)$. By compute a basis I mean, finding the beginning of the Fourier ...