# Questions tagged [half-integral-weight]

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

21 questions
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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 ...
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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 ...
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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 $F(m) = m^2$ ...
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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 ...
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### 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 $L$-...
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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, $\kappa \in \frac 12 \mathbb{Z} - \... 1answer 620 views ### The space of lattices and modular forms of weight 1/2 Suppose my favorite way of thinking about modular forms is as functions on the space of (real, 2D) lattices. One can identify this space with$SL_2(\mathbb{Z}) \backslash GL_2(\mathbb{R})$, i.e. ... 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$...
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 ...