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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_{...
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 ...
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 ...
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 ...
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} - \...