The Habiro ring of a number field
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915464772321280 |
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| author | Garoufalidis, Stavros Scholze, Peter Wheeler, Campbell Zagier, Don |
| author_facet | Garoufalidis, Stavros Scholze, Peter Wheeler, Campbell Zagier, Don |
| contents | We introduce the Habiro ring of a number field $\mathbb{K}$ and modules over it graded by $K_3(\mathbb{K})$. Elements of these modules are collections of power series at each complex root of unity that arithmetically glue with each other after applying a Frobenius endomorphism, and after dividing at each prime by a collection of series that depends solely on an element of the Bloch group.
The main theorems of this paper concern number fields, their algebraic $K$-theory and its regulator maps (Borel, $p$-adic and étale), whereas the explicit collections of series are defined by a careful algebraic analysis of the infinite Pochhammer symbol at roots of unity.
The origin of the above mentioned power series comes from perturbative Chern--Simons theory and by expansions of the admissible series of Kontsevich--Soibelman, both ultimately related to the infinite Pochhammer symbol. This link suggests that some Donaldson-Thomas invariants have arithmetic meaning and that some elements of the Habiro ring of a number field have enumerative meaning.
Added subsection 1.1 explaining what the paper is about and subsection 1.8 explaining the relation to perturbative complex Chern-Simons theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Habiro ring of a number field Garoufalidis, Stavros Scholze, Peter Wheeler, Campbell Zagier, Don Number Theory High Energy Physics - Theory Geometric Topology We introduce the Habiro ring of a number field $\mathbb{K}$ and modules over it graded by $K_3(\mathbb{K})$. Elements of these modules are collections of power series at each complex root of unity that arithmetically glue with each other after applying a Frobenius endomorphism, and after dividing at each prime by a collection of series that depends solely on an element of the Bloch group. The main theorems of this paper concern number fields, their algebraic $K$-theory and its regulator maps (Borel, $p$-adic and étale), whereas the explicit collections of series are defined by a careful algebraic analysis of the infinite Pochhammer symbol at roots of unity. The origin of the above mentioned power series comes from perturbative Chern--Simons theory and by expansions of the admissible series of Kontsevich--Soibelman, both ultimately related to the infinite Pochhammer symbol. This link suggests that some Donaldson-Thomas invariants have arithmetic meaning and that some elements of the Habiro ring of a number field have enumerative meaning. Added subsection 1.1 explaining what the paper is about and subsection 1.8 explaining the relation to perturbative complex Chern-Simons theory. |
| title | The Habiro ring of a number field |
| topic | Number Theory High Energy Physics - Theory Geometric Topology |
| url | https://arxiv.org/abs/2412.04241 |