Linking numbers and non-holomorphic Siegel modular forms

TL;DR

This paper investigates the relationship between linking numbers in hyperbolic 3-manifolds and genus 2 Siegel modular forms, revealing new modular properties and bounds for linking numbers.

Contribution

It establishes that generating series of linking numbers converge to functions with genus 2 Siegel modular form transformation properties, connecting topology and automorphic forms.

Findings

Generated series converge to genus 2 Siegel modular forms

Explicit modifications exhibit modular transformation properties

Linking numbers are polynomially bounded

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.