The exotic inverted Kloosterman sum
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915048782299136 |
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| author | Fu, Lei Wan, Daqing |
| author_facet | Fu, Lei Wan, Daqing |
| contents | Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $ψ:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $χ: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}} χ(x)ψ\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1 = \dim_{\mathbb F_q}B$. Up to a trivial main term, this gives the expected square root cancellation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The exotic inverted Kloosterman sum Fu, Lei Wan, Daqing Number Theory 14F20, 11L05 Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $ψ:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $χ: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}} χ(x)ψ\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1 = \dim_{\mathbb F_q}B$. Up to a trivial main term, this gives the expected square root cancellation. |
| title | The exotic inverted Kloosterman sum |
| topic | Number Theory 14F20, 11L05 |
| url | https://arxiv.org/abs/2407.03599 |