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Resonances and \(\Omega\)-results for exponential sums related to Maass forms for \(\mathrm{SL}(n, \mathbb{Z})\). (English) Zbl 1357.11067
The \(\varphi\) be a \(\mathrm{GL}(n)\) automorphic form that is invariant under \(\mathrm{SL}(n,{\mathbb Z})\), and \(a(m_1,\cdots,m_{n-1})\) denote its Fourier coefficients. Then the standard \(L\)-function of \(\varphi\) is given for \(\mathrm{Re}(s)\gg0\) by \(L(s,\varphi)=\sum_{n\geq1} a(m,1,\dots,1) m^{-s}\). In this paper the authors obtain resonances for short exponential sums weighted by the Fourier coefficients \(a(m,1,\dots,1)\). Their main result is an estimate for \[ \sum_{M\leq m\leq M+\Delta} a(m,1,\dots,1) w(m) \exp(d^{1/n} m /M^{1-1/n}) \] for a suitable weight function \(w\) supported in \([M,M+\Delta]\). Here \(M^{1-1/n+\varepsilon}\ll \Delta\ll M\) and \(d\) is a fixed positive integer. To do so they derive asymptotics for integrals appearing in the (untwisted) \(\mathrm{GL}(n)\) Voronoi summation formula. As an application, they prove an \(\Omega\)-result for short unweighted sums of these Fourier coefficients, that is, for \(\sum_{M\leq m\leq M+\Delta} a(m,1,\dots,1)\). And as another consequence, they show that given any sequence of \(M_1,M_2,\dots\) of positive real numbers tending to infinity and any \(\varepsilon>0\), the coefficients \(a(m,1,\dots,1)\) for \(m\) in the union of intervals \([M_\ell,M_\ell+M_\ell^{1-1/n+\varepsilon}]\) uniquely determine the automorphic form \(\varphi\).

11L03 Trigonometric and exponential sums, general
11F30 Fourier coefficients of automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
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