Mellin transform formulas for Drinfeld modules
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917049558630400 |
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| author | Gezmiş, Oğuz Green, Nathan |
| author_facet | Gezmiş, Oğuz Green, Nathan |
| contents | We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of $t$-motives. We then specialize our formulas to express special values of Goss $L$-functions as Drinfeld periods multiplied by rigid analytic trivializations evaluated under this motivic map. We view these formulas as characteristic-$p$ analogues of integral representations of Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld modules tensored with the tensor powers of the Carlitz module, which serves as the Tate twist of a Drinfeld module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mellin transform formulas for Drinfeld modules Gezmiş, Oğuz Green, Nathan Number Theory 11G09 (Primary), 11M38 (Secondary) We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of $t$-motives. We then specialize our formulas to express special values of Goss $L$-functions as Drinfeld periods multiplied by rigid analytic trivializations evaluated under this motivic map. We view these formulas as characteristic-$p$ analogues of integral representations of Hasse-Weil type zeta functions. We also apply this machinery for Drinfeld modules tensored with the tensor powers of the Carlitz module, which serves as the Tate twist of a Drinfeld module. |
| title | Mellin transform formulas for Drinfeld modules |
| topic | Number Theory 11G09 (Primary), 11M38 (Secondary) |
| url | https://arxiv.org/abs/2405.02915 |