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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 $$\varphi$$.

##### MSC:
 11L03 Trigonometric and exponential sums, general 11F30 Fourier coefficients of automorphic forms 11F37 Forms of half-integer weight; nonholomorphic modular forms
GL(n)pack
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