Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912597212659712 |
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| author | Leung, Wing Hong |
| author_facet | Leung, Wing Hong |
| contents | 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 $H\subset[\![ -X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and $\{a(h)\}_{h\in H}\subset\mathbb{C}$ be a sequence. We show that if $H\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$, \begin{align*}
D_{a,H}(X):=\frac{1}{|H|}\sum_{h\in 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}}{|H|}\|a\|_2
\end{align*}
for any $\varepsilon>0$, and a similar bound when $|H|$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_13081 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets Leung, Wing Hong Number Theory 11F30, 11M41 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 $H\subset[\![ -X^{1-\varepsilon},X^{1+\varepsilon}]\!]$ and $\{a(h)\}_{h\in H}\subset\mathbb{C}$ be a sequence. We show that if $H\subset \ell+[\![ 0,X^{1/2+\varepsilon}]\!] $ for some $\ell\geq0$, \begin{align*} D_{a,H}(X):=\frac{1}{|H|}\sum_{h\in 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}}{|H|}\|a\|_2 \end{align*} for any $\varepsilon>0$, and a similar bound when $|H|$ 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. |
| title | Shifted Convolution Sums for $GL(3)\times GL(2)$ Averaged over weighted sets |
| topic | Number Theory 11F30, 11M41 |
| url | https://arxiv.org/abs/2210.13081 |