论文标题
双方平方矩和共振总和的边界
Double square moments and bounds for resonance sums of cusp forms
论文作者
论文摘要
令$ f $和$ g $为模块化组的holomorphic cusp表格$ sl_2（\ mathbb z）的重量$ k_1 $和$ k_2 $，分别具有傅立叶系数$λ_f（n）$和$λ_g（n）$。对于真实的$α\ neq0 $和$ 0 <β\ leq1 $，请考虑$λ_f（n）λ_g（n）λ_g（n）$的平滑共振总和$ s_x（f，g;α，β）$，涉及$ e（αn^β）$ x \ leq n \ leq n \ leq2x $。 $ f $和$ g $的双方平方矩$ s_x（f，g;α，β）$当它们的重量$ k_1 $和$ k_2 $倾向于将无限限制在一起。通过允许$ f $和$ g $移动，这些双重矩确实是与$ gl（4）$的自动形态形式相关的平方米。通过取出一小部分$ f $和$ g $的特殊集合，将证明单个$ s_x（f，g;α，β）$的界限。这些个别界限打破了$ x^\ frac58 $，对于$ \ frac16 <β<1 $，并实现了几乎所有$ f $和$ g $的$ \ frac13 <β<1 $的平方根取消，作为Cusp cusp yntegers cusp byesteps of Integers的证据。这项研究中使用的方法包括彼得森的公式，泊松的总和公式和固定相积分。
Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. For real $α\neq0$ and $0<β\leq1$, consider a smooth resonance sum $S_X(f,g;α,β)$ of $λ_f(n)λ_g(n)$ against $e(αn^β)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;α,β)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking out a small exceptional set of $f$ and $g$, bounds for individual $S_X(f,g;α,β)$ will then be proved. These individual bounds break the resonance barrier of $X^\frac58$ for $\frac16<β<1$ and achieve a square-root cancellation for $\frac13<β<1$ for almost all $f$ and $g$ as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.
