论文标题
$ gl（3）\ times gl（2）$平均在加权集上的转移卷积总和
Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets
论文作者
论文摘要
令$ a（1，m）$为$ sl（3，\ m athbb {z}）$ hecke-maass cusp form $π_1$和$λ（m）$的傅立叶系数是$ sl（2，2，\ mathbb {z}）令$ \ mathcal {h} \ subset [\！[-x^{1- \ varepsilon}，x^{1+ \ varepsilon}] \！] \！] $和$ \ {a（h）\} _（h）_ {我们表明，如果$ \ Mathcal {h} \ subset \ ell+[\！[0，x^{1/2+\ varepsilon}] \！] $ for Some $ \ ell \ geq0 $，\ begin {align*} d_ {a，\ Mathcal {h}}（x）：= \ frac {1} {| \ Mathcal {h} |} \ sum_ {h \ in \ nathcal {h}} a（h}} a（h） a（1，m）λ（rm+h）v \ left（\ frac {m} {x} \ right）\ ll_ {π_1，π_2，π_2，\ varepsilon} \ frac {x^^{x^{1+ \ varepsilon}} {1+ \ varepsilon}}} { \ end {align*} 对于任何$ \ varepsilon> 0 $，并且当$ | \ Mathcal {h} | $很大时，类似的绑定。这可以改善Sun的界限，并以任意权重的平均值将其推广到平均值。此外，我们演示了如何通过研究加权平均值的移位总和来恢复Jutila圆方法给出的可分解模量结构。这使我们能够在不使用Jutila圆方法的情况下以固定的偏移来恢复Munshi的界限。
Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $π_1$ and $λ(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $π_2$. Let $\mathcal{H}\subset[\![ -X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and $\{a(h)\}_{h\in\mathcal{H}}\subset\mathbb{C}$ be a sequence. We show that if $\mathcal{H}\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$, \begin{align*} D_{a,\mathcal{H}}(X):=\frac{1}{|\mathcal{H}|}\sum_{h\in\mathcal{H}}a(h)\sum_{m=1}^\infty A(1,m)λ(rm+h)V\left(\frac{m}{X}\right)\ll_{π_1,π_2,\varepsilon} \frac{X^{1+\varepsilon}}{|\mathcal{H}|}\|a\|_2 \end{align*} for any $\varepsilon>0$, and a similar bound when $|\mathcal{H}|$ is big. This improves Sun's bound and generalizes it to an average with arbitrary weights. Moreover, we demonstrate how one can recover the factorizable moduli structure given by the Jutila's circle method via studying a shifted sum with weighted average. This allows us to recover Munshi's bound on the shifted sum with a fixed shift without using the Jutila's circle method.
