Linking numbers and non-holomorphic Siegel modular forms

Mads Bjerge Christensen

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arXiv · 2410.17231

Linking numbers and non-holomorphic Siegel modular forms

Mads Bjerge Christensen

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Abstract

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.

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Mads Bjerge Christensen. 2024-10-22. Linking numbers and non-holomorphic Siegel modular forms. https://arxiv.org/abs/2410.17231

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