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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2212.07823 |
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| _version_ | 1866910443184848896 |
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| author | Ferraro, Giacomo Hermes |
| author_facet | Ferraro, Giacomo Hermes |
| contents | Goss zeta values can be found, in some cases, as evaluations of a new type of rigid analytic function on projective curves $X$ over a finite field $\mathbb{F}_q$, called "Pellarin $L$-series". In the case of genus $0$ and $1$, Pellarin and Green--Papanikolas further determined functional identities for Pellarin $L$-series, in partial analogy with the functional equation of Dirichlet $L$-series.
The aim of this paper is to prove that a generalization of these functional identities holds in arbitrary genus. Our proof exploits the topological nature of divisors on the curve $X$, as well as the introduction of an "adjoint shtuka function". This allows us to reinterpret Pellarin $L$-series as dual versions of the special functions studied by Anglès, Ngo Dac, and Tavares Ribeiro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07823 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A class of functional identities associated to curves over finite fields Ferraro, Giacomo Hermes Number Theory 11G09 Goss zeta values can be found, in some cases, as evaluations of a new type of rigid analytic function on projective curves $X$ over a finite field $\mathbb{F}_q$, called "Pellarin $L$-series". In the case of genus $0$ and $1$, Pellarin and Green--Papanikolas further determined functional identities for Pellarin $L$-series, in partial analogy with the functional equation of Dirichlet $L$-series. The aim of this paper is to prove that a generalization of these functional identities holds in arbitrary genus. Our proof exploits the topological nature of divisors on the curve $X$, as well as the introduction of an "adjoint shtuka function". This allows us to reinterpret Pellarin $L$-series as dual versions of the special functions studied by Anglès, Ngo Dac, and Tavares Ribeiro. |
| title | A class of functional identities associated to curves over finite fields |
| topic | Number Theory 11G09 |
| url | https://arxiv.org/abs/2212.07823 |