On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909273929285632 |
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| author | Gezmiş, Oğuz Namoijam, Changningphaabi |
| author_facet | Gezmiş, Oğuz Namoijam, Changningphaabi |
| contents | Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(θ)$. For a Drinfeld module $ϕ$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_ϕ$ of $ϕ$. Moreover, when $ϕ$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_ϕ$ at any positive integer is transcendental over $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_02569 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules Gezmiş, Oğuz Namoijam, Changningphaabi Number Theory 11G09, 11M38, 11J93 Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the rational function field $K:=\mathbb{F}_q(θ)$. For a Drinfeld module $ϕ$ defined over $K$, we study the transcendence of special values of the Goss $L$-function attached to the abelian $t$-motive $M_ϕ$ of $ϕ$. Moreover, when $ϕ$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has everywhere good reduction, we prove that the value of the Goss $L$-function attached to the $(r-1)$-st exterior power of $M_ϕ$ at any positive integer is transcendental over $K$. |
| title | On the transcendence of special values of Goss $L$-functions attached to Drinfeld modules |
| topic | Number Theory 11G09, 11M38, 11J93 |
| url | https://arxiv.org/abs/2110.02569 |