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3 votes
0 answers
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 ...
davidlowryduda's user avatar
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-...
BHT's user avatar
  • 191
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 ...
Henri Cohen's user avatar
  • 13.1k
1 vote
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 ...
SUNIL PASUPULATI's user avatar
5 votes
1 answer
344 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_{...
BH NT's user avatar
  • 135
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(\...
Stabilo's user avatar
  • 1,479
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}\...
Med's user avatar
  • 400
3 votes
1 answer
344 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}{\...
user95750's user avatar
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\...
Med's user avatar
  • 400
8 votes
1 answer
628 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 \...
user129098's user avatar
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 ...
Fan Zheng's user avatar
  • 5,169
5 votes
2 answers
466 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 ...
Thomas Hulse's user avatar
1 vote
1 answer
324 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 ...
Eren Mehmet Kiral's user avatar
9 votes
1 answer
748 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 $F(m) = m^2$ ...
stl's user avatar
  • 585
19 votes
3 answers
1k 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 ...
Valerie's user avatar
  • 955
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 $L$-...
N. Kumar's user avatar
  • 592
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} - \...
Eren Mehmet Kiral's user avatar
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. ...
Freddie Manners's user avatar
5 votes
1 answer
718 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 ...
Underflow's user avatar