Analog of theta-lifting for a curve over dual numbers over a finite field
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913909544321024 |
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| author | Kazhdan, David Polishchuk, Alexander |
| author_facet | Kazhdan, David Polishchuk, Alexander |
| contents | We continue the study of automorphic functions associated with a curve $C$ over the ring $k[ε]/(ε^2)$, where $k$ is a finite field, begun in arXiv:2303.16259. Namely, we study an example of theta-lifting in this framework and show that it can be understood in terms of the orbit decomposition of the space of automorphic functions $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ introduced in loc.cit. We prove that all strongly cuspidal functions in $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ can be constructed using theta-lifting for an appropriate double covering $\tilde{C}\to C$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_19036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analog of theta-lifting for a curve over dual numbers over a finite field Kazhdan, David Polishchuk, Alexander Algebraic Geometry Number Theory We continue the study of automorphic functions associated with a curve $C$ over the ring $k[ε]/(ε^2)$, where $k$ is a finite field, begun in arXiv:2303.16259. Namely, we study an example of theta-lifting in this framework and show that it can be understood in terms of the orbit decomposition of the space of automorphic functions $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ introduced in loc.cit. We prove that all strongly cuspidal functions in $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ can be constructed using theta-lifting for an appropriate double covering $\tilde{C}\to C$. |
| title | Analog of theta-lifting for a curve over dual numbers over a finite field |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2506.19036 |