Questions tagged [half-integral-weight]
Questions about half-integral weight modular forms, and more generally automorphic representations associated to metaplectic groups.
27 questions
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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 ...
- 955
16
votes
3
answers
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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 $...
- 6,290
14
votes
3
answers
2k
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 $L$-...
- 592
12
votes
1
answer
874
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. ...
- 915
9
votes
1
answer
748
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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 $F(m) = m^2$ ...
- 585
8
votes
4
answers
3k
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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 ...
- 2,810
8
votes
1
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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 $j \ge 1$.
$$
a(N) := \frac1j \left(m \...
- 83
8
votes
1
answer
739
views
Average of Fourier coefficients of a cusp form of half integral weight
Suppose $f$ is a cusp form of half integral weight $k$ w.r.t. the group $\Gamma_0(4)$ ($k$ is not very low, can assume $k \ge 11/2$), and $a_n$ is its Fourier coefficient. The Linnik bound says that ...
- 5,169
6
votes
1
answer
480
views
Twisted modular forms of half-integral weight
I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of
$$ \theta(\...
- 1,479
6
votes
1
answer
497
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} - \...
6
votes
0
answers
174
views
Half-integral weight slash operator
I am aware that the correct way to look at modular forms of half-integral weight is on the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$. Assume however that we insist on considering them as ...
- 13.1k
5
votes
1
answer
344
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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_{...
- 135
5
votes
1
answer
718
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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 ...
- 51
5
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1
answer
545
views
(Explicit) Basis for Kohnen's plus-space of modular forms of half integral weight
Sorry if this is trivial, but I could not find any reference.
Let $k,a,b$ be integers. The space of modular forms of integer weight $M_k(\text{SL}_2(\mathbb{Z}))$ admits a basis of the form $\{ E_4^...
- 367
5
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2
answers
466
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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 ...
- 53
4
votes
1
answer
184
views
Necessity of conditions $N$ odd, square-free and $\chi$ quadratic in Kohnen's plus space - modular forms of half-integral weight
Kohnen introduced the "plus" space as a subspace of the space of modular forms of half integral weight, first in his 1980 paper and then generalized the work in a later 1982 paper. Why is ...
- 367
4
votes
0
answers
187
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Understanding Shimura correspondence in context of Langlands functoriality
Recently, I started to read about automorphic forms and representations on covering groups, e.g. metaplectic groups. I set my first goal as understanding Shimura's correspondence in representation ...
- 2,215
3
votes
1
answer
344
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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:
$$X\ll \dfrac{\left(\sum_{n\le X}a(n)\right)^2}{\...
- 39
3
votes
1
answer
238
views
Half integral weight modular forms that reduce to a nonzero constant modulo a given prime
Let $B_k = \frac{N_k}{D_k}$ be the reduced numerator and denominator of the $k$-th Bernoulli number. For a given prime $p>2$, the (unconventionally normalized) Eisenstein series $E_{p-1}(z) = N_{p-...
- 191
3
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0
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79
views
Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form
In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
- 1,570
3
votes
0
answers
235
views
Functional equation link two Dirichlet series
Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let
$$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and
$$R_f(s)=\sum_{n\ge 1}\...
- 400
3
votes
0
answers
362
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 ...
- 334
2
votes
0
answers
212
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}...
- 11.2k
1
vote
1
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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 ...
1
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0
answers
354
views
What is definition of Cohen–Eisenstein series?
I only find Cohen–Eisenstein series of weight 3/2 (in the paper A note on the Fourier coefficients of a Cohen-Eisenstein series). I founded some general definition in "Modular Forms with Integral and ...
- 743
1
vote
0
answers
134
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 ...
0
votes
1
answer
218
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\...
- 400