Zagier-Hoffman's conjectures in positive characteristic II
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916130079113216 |
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| author | Im, Bo-Hae Kim, Hojin Le, Khac Nhuan Dac, Tuan Ngo Pham, Lan Huong |
| author_facet | Im, Bo-Hae Kim, Hojin Le, Khac Nhuan Dac, Tuan Ngo Pham, Lan Huong |
| contents | Zagier-Hoffman's conjectures predict the dimension and a basis for the $\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values (MZV's) of fixed weight where $N$ is a natural number.
For $N=1$ (MZV's case), half of these conjectures have been solved by the work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's identity. The other half are completely open. For $N=2$ (alternating MZV's case) and $N=3,4,8$, Deligne-Goncharov and Deligne solved the same half of these conjectures for $N$th-cyclotomic MZV's. For other values of $N$, no sharp upper bound on the dimension is known.
In this paper we completely establish, for all $N$, Zagier-Hoffman's conjectures for $N$th cyclotomic multiple zeta values in positive characteristic. By working with the tower of all cyclotomic extensions, we present a proof that is uniform on $N$ and give an effective algorithm to express any cyclotomic multiple zeta value in the chosen basis. This generalizes all previous work on these conjectures for MZV's and alternating MZV's in positive characteristic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_11539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zagier-Hoffman's conjectures in positive characteristic II Im, Bo-Hae Kim, Hojin Le, Khac Nhuan Dac, Tuan Ngo Pham, Lan Huong Number Theory Primary 11M32, Secondary 11G09, 11J93, 11M38, 11R58 Zagier-Hoffman's conjectures predict the dimension and a basis for the $\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values (MZV's) of fixed weight where $N$ is a natural number. For $N=1$ (MZV's case), half of these conjectures have been solved by the work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's identity. The other half are completely open. For $N=2$ (alternating MZV's case) and $N=3,4,8$, Deligne-Goncharov and Deligne solved the same half of these conjectures for $N$th-cyclotomic MZV's. For other values of $N$, no sharp upper bound on the dimension is known. In this paper we completely establish, for all $N$, Zagier-Hoffman's conjectures for $N$th cyclotomic multiple zeta values in positive characteristic. By working with the tower of all cyclotomic extensions, we present a proof that is uniform on $N$ and give an effective algorithm to express any cyclotomic multiple zeta value in the chosen basis. This generalizes all previous work on these conjectures for MZV's and alternating MZV's in positive characteristic. |
| title | Zagier-Hoffman's conjectures in positive characteristic II |
| topic | Number Theory Primary 11M32, Secondary 11G09, 11J93, 11M38, 11R58 |
| url | https://arxiv.org/abs/2402.11539 |