Average shifted convolution sum for $GL(d_1)\times GL(d_2)$
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arXiv
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| Format: | Preprint |
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2025
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| author | Molla, Esrafil Ali |
| author_facet | Molla, Esrafil Ali |
| contents | We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{π_1}(n)\, A_{π_2}(n+h), $$ where $A_{π_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $π_i$ for $\mathrm{SL}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_23096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Average shifted convolution sum for $GL(d_1)\times GL(d_2)$ Molla, Esrafil Ali Number Theory Primary: 11F30, 11F66, Secondary: 11M41 We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{π_1}(n)\, A_{π_2}(n+h), $$ where $A_{π_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $π_i$ for $\mathrm{SL}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect. |
| title | Average shifted convolution sum for $GL(d_1)\times GL(d_2)$ |
| topic | Number Theory Primary: 11F30, 11F66, Secondary: 11M41 |
| url | https://arxiv.org/abs/2511.23096 |