The sup-norm problem beyond the newform. (English) Zbl 07836149
Take an irreducible automorphic representation \(\pi\) of \(\mathrm{GL}_2\) over the adele ring over \({\mathbb Q}\), let \(p>3\) be a prime and assume \(\pi\) is unramified away from \(p\). Let \(m\) be a large integer, and \(V\) be the space of automorphic forms in \(\pi\) of level \(p^m\). With the extra assumption that \(\pi\) is twist-minimal, the paper establishes a nontrivial bound regarding the sup-norm of
\[
\Phi(g)=(\sum_{i=1}^d \vert \phi_i(g)\vert^2)^{\frac12},
\]
where \(d\) is the dimension of \(V\) and \(\{\phi_i\}\) is an orthonormal basis of \(V\).
The main result states that \[ \|\Phi\|_{\infty}\ll T^{\frac12+\varepsilon}d^{\frac{11}{12}+\varepsilon} \] where \(T=1+|t_\pi|\) with \(t_\pi\) the spectral parameter of \(\pi\). This is to be compared with the trivial bound of \(T^{\frac12+\varepsilon}d\). The gain is in the dimension aspect. Note that as \(m\) increases, \(d\) goes to infinity.
The main result states that \[ \|\Phi\|_{\infty}\ll T^{\frac12+\varepsilon}d^{\frac{11}{12}+\varepsilon} \] where \(T=1+|t_\pi|\) with \(t_\pi\) the spectral parameter of \(\pi\). This is to be compared with the trivial bound of \(T^{\frac12+\varepsilon}d\). The gain is in the dimension aspect. Note that as \(m\) increases, \(d\) goes to infinity.
Reviewer: Zhengyu Mao (Newark)
MSC:
| 11F03 | Modular and automorphic functions |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F85 | \(p\)-adic theory, local fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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