Modular Symbols with Values in Beilinson-Kato Distributions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911099469692928 |
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| author | Busuioc, Cecilia Park, Jeehoon Patashnick, Owen Stevens, Glenn |
| author_facet | Busuioc, Cecilia Park, Jeehoon Patashnick, Owen Stevens, Glenn |
| contents | For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bmξ_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field.
The Siegel distribution $\bmμ$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bmξ_n$; we define $\bmξ_n$ essentially by taking the $n$-Steinberg product of $\bmμ$.
The most non-trivial part of this construction is the cocycle property of $\bmξ_n$; we prove it by using an induction on $n$ based on the first two cases $\bmξ_1$ and $\bmξ_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_14620 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Modular Symbols with Values in Beilinson-Kato Distributions Busuioc, Cecilia Park, Jeehoon Patashnick, Owen Stevens, Glenn Number Theory Algebraic Geometry K-Theory and Homology 14F67 (Primary) 19D45 (Secondary) For each integer $n\geq 1$, we construct a $\operatorname{GL}_n(\mathbb Q)$-invariant modular symbol $\bmξ_n$ with coefficients in a space of distributions that takes values in the Milnor $K_n$-group of the modular function field. The Siegel distribution $\bmμ$ on $\mathbb Q^2$, with values in the modular function field, serves as the building block for $\bmξ_n$; we define $\bmξ_n$ essentially by taking the $n$-Steinberg product of $\bmμ$. The most non-trivial part of this construction is the cocycle property of $\bmξ_n$; we prove it by using an induction on $n$ based on the first two cases $\bmξ_1$ and $\bmξ_2$; the first case is trivial, and the second case essentially follows from the fact that Beilinson-Kato elements in the Milnor $K_2$-group modulo torsion satisfy the Manin relations. |
| title | Modular Symbols with Values in Beilinson-Kato Distributions |
| topic | Number Theory Algebraic Geometry K-Theory and Homology 14F67 (Primary) 19D45 (Secondary) |
| url | https://arxiv.org/abs/2311.14620 |