An extension of Deligne-Henniart's twisting formula and its applications
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911757014925312 |
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| author | Biswas, Sazzad Ali |
| author_facet | Biswas, Sazzad Ali |
| contents | Let $F/\bbQ_p$ be a non-Archimedean local field, and $G_F$ be the absolute Galois group of $F$. Let $ρ_1$ and $ρ_2$ be two finite-dimensional complex representations of $G_F$. Let $ψ$ be a nontrivial additive character of $F$. Then, the question is:
What is the twisting formula for the root number $W(ρ_1\otimesρ_2,ψ)$?}
In general, the answer to this question is not yet known. However, if one of $ρ_i \quad(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for $W(ρ_1\otimesρ_2,ψ)$. Later, in [12], Deligne and Henniart gave a general twisting formula for a {\it zero}-dimensional virtual representation twisted by a finite-dimensional representation of $G_F$. In this paper, we first extend Deligne's twisting formula for U-isotropic Heisenberg representation of dimension prime $p$, then we further extend Deligne-Henniart's result.
Finally, we provide two very important applications of our twisting formula: --
(i) invariant formula for the local root numbers for U-isotropic Heisenberg representations, and (ii) a converse theorem on the Galois side. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_08118 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | An extension of Deligne-Henniart's twisting formula and its applications Biswas, Sazzad Ali Number Theory Let $F/\bbQ_p$ be a non-Archimedean local field, and $G_F$ be the absolute Galois group of $F$. Let $ρ_1$ and $ρ_2$ be two finite-dimensional complex representations of $G_F$. Let $ψ$ be a nontrivial additive character of $F$. Then, the question is: What is the twisting formula for the root number $W(ρ_1\otimesρ_2,ψ)$?} In general, the answer to this question is not yet known. However, if one of $ρ_i \quad(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor, then in [13], Deligne gave a twisting formula for $W(ρ_1\otimesρ_2,ψ)$. Later, in [12], Deligne and Henniart gave a general twisting formula for a {\it zero}-dimensional virtual representation twisted by a finite-dimensional representation of $G_F$. In this paper, we first extend Deligne's twisting formula for U-isotropic Heisenberg representation of dimension prime $p$, then we further extend Deligne-Henniart's result. Finally, we provide two very important applications of our twisting formula: -- (i) invariant formula for the local root numbers for U-isotropic Heisenberg representations, and (ii) a converse theorem on the Galois side. |
| title | An extension of Deligne-Henniart's twisting formula and its applications |
| topic | Number Theory |
| url | https://arxiv.org/abs/2101.08118 |