Linking numbers and non-holomorphic Siegel modular forms
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909585322803200 |
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| author | Christensen, Mads Bjerge |
| author_facet | Christensen, Mads Bjerge |
| contents | We study generating series encoding linking numbers between geodesics in arithmetic hyperbolic $3$-folds. We show that the series converge to functions on genus $2$ Siegel space and that certain explicit modifications have the transformation properties of genus $2$ Siegel modular forms of weight $2$. This is done by carefully analyzing the integral of the Kudla--Millson theta series over a Seifert surface with geodesic boundary. As a corollary, we deduce a polynomial bound on the linking numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17231 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linking numbers and non-holomorphic Siegel modular forms Christensen, Mads Bjerge Number Theory Differential Geometry Geometric Topology We study generating series encoding linking numbers between geodesics in arithmetic hyperbolic $3$-folds. We show that the series converge to functions on genus $2$ Siegel space and that certain explicit modifications have the transformation properties of genus $2$ Siegel modular forms of weight $2$. This is done by carefully analyzing the integral of the Kudla--Millson theta series over a Seifert surface with geodesic boundary. As a corollary, we deduce a polynomial bound on the linking numbers. |
| title | Linking numbers and non-holomorphic Siegel modular forms |
| topic | Number Theory Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2410.17231 |