论文标题
$ l^p $ - 自动形式通过光谱互惠的界限
$L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity
论文作者
论文摘要
令$ g $为模块化表面$ {\ rm sl} _2（\ Mathbb {z}）\ Backslash \ Mathbb {h} $的Hecke-Maass尖缘形式，即SL} _2（\ Mathbb {Z}）\ BackSlash \ Mathbb {H} $，是每个Hecke操作员的联合特征功能。我们证明了$ l^4 $ -norm绑定$ \ | g \ | _4 \ ll _ {\ varepsilon}λ_g^{3/304+\ varepsilon} $，其中$λ_g$表示laplacian eigenvalue of $ g $，这改善$ \ | g \ | _4 \llλ_g^{1/16} $对于紧凑型Riemann表面上的Laplacian eigenfunctions，超过六倍以上的功率保存。通过插值，这可以为Hecke-Maass cusp表单提供$ l^p $ -NORM的界限，这些表单对Sogge的所有$ p> 2 $进行了节省的改进。我们的论文标志着Sogge在模块化表面上的结果的首次改进。此外，这些方法的紧凑型算术表面最佳的$ l^4 $ - norm截至迄今为止。 通过Watson-Ichino三重产品公式，对于$ L $ functions的某些混合时刻，$ l^4 $ norm of $ g $的界限将减少到边界。我们使用两种形式的光谱互惠束缚这些。首先是$ {\ rm gl} _3 \ times {\ rm gl} _2 \ leftrightsquigarrow {\ rm gl} _4 \ times {\ rm gl} _4 gl} _3 \ times {\ rm gl} _2 $ rankin-selberg $ l $ l $ - for $ {\ rm gl} _1 $时刻$ {\ rm gl} _4 \ times {\ rm gl} _4 \ times {\ rm gl} _1 _1 $ rankin-selberg $ l $ l $ -functions;这可以看作是Motohashi公式的尖锐类似物，将Riemann Zeta函数的第四刻与Hecke $ l $ functions的中心值的第三刻有关。第二个形式是$ {\ rm gl} _4 \ times {\ rm gl} _2 \ leftrightsquigarrow {\ rm gl} _4 \ times {\ rm gl} _2 _2 $ spectralcity spectral choppitral，这是一个cuspidal of for four keuznetsov of keuznetsov of Kuznetsov of Kuznetsov of Kuznetsov $ L $ - 功能。
Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a joint eigenfunction of every Hecke operator. We prove the $L^4$-norm bound $\|g\|_4\ll_{\varepsilon}λ_g^{3/304+\varepsilon}$, where $λ_g$ denotes the Laplacian eigenvalue of $g$, which improves upon Sogge's $L^4$-norm bound $\|g\|_4\llλ_g^{1/16}$ for Laplacian eigenfunctions on a compact Riemann surface by more than a six-fold power-saving. Via interpolation, this yields $L^p$-norm bounds for Hecke-Maass cusp forms that are power-saving improvements on Sogge's bounds for all $p>2$. Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best $L^4$-norm bound to date. Via the Watson-Ichino triple product formula, bounds for the $L^4$-norm of $g$ are reduced to bounds for certain mixed moments of $L$-functions. We bound these using two forms of spectral reciprocity. The first is a form of ${\rm GL}_3\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_1$ spectral reciprocity, which relates a ${\rm GL}_2$ moment of ${\rm GL}_3\times{\rm GL}_2$ Rankin-Selberg $L$-functions to a ${\rm GL}_1$ moment of ${\rm GL}_4\times{\rm GL}_1$ Rankin-Selberg $L$-functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke $L$-functions. The second is a form of ${\rm GL}_4\times{\rm GL}_2\leftrightsquigarrow{\rm GL}_4\times{\rm GL}_2$ spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke $L$-functions.
