Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914452989804544 |
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| author | Ahlquist, Victor |
| author_facet | Ahlquist, Victor |
| contents | We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point. This generalises and extends results over $\mathbb{Q}$ due to Durlanik, proving that an infinite number of these $L$-functions are non-vanishing.
We obtain these results by examining the low-lying zeros of the $L$-functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet $L$-functions associated with quadratic function field extensions, to the $D_4$-case. The main difficulty is studying $L$-functions which are associated to $D_4$-fields whose quadratic subfield is of large discriminant. These $L$-functions are studied by utilising the so-called flipped field of a $D_4$ extension, combining a method introduced by Friedrichsen for counting $D_4$-fields, with explicit ramification theory in such fields provided by Altuğ, Shankar, Varma and Wilson. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14576 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor Ahlquist, Victor Number Theory 11R16, 11R45, 11R59 (Primary) 11M50 (Secondary) We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point. This generalises and extends results over $\mathbb{Q}$ due to Durlanik, proving that an infinite number of these $L$-functions are non-vanishing. We obtain these results by examining the low-lying zeros of the $L$-functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet $L$-functions associated with quadratic function field extensions, to the $D_4$-case. The main difficulty is studying $L$-functions which are associated to $D_4$-fields whose quadratic subfield is of large discriminant. These $L$-functions are studied by utilising the so-called flipped field of a $D_4$ extension, combining a method introduced by Friedrichsen for counting $D_4$-fields, with explicit ramification theory in such fields provided by Altuğ, Shankar, Varma and Wilson. |
| title | Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor |
| topic | Number Theory 11R16, 11R45, 11R59 (Primary) 11M50 (Secondary) |
| url | https://arxiv.org/abs/2511.14576 |