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Linking numbers and non-holomorphic Siegel modular forms

Mads Bjerge Christensen

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.

Linking numbers and non-holomorphic Siegel modular forms

Abstract

We study generating series encoding linking numbers between geodesics in arithmetic hyperbolic -folds. We show that the series converge to functions on genus Siegel space and that certain explicit modifications have the transformation properties of genus Siegel modular forms of weight . 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.

Paper Structure

This paper contains 28 sections, 37 theorems, 226 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Hyperbolic geodesics
  3. Main results
  4. Ideas of proof
  5. Geodesic linking numbers
  6. Hyperbolic 3-folds
  7. Geodesic cycles
  8. The forms f0(X) and y0(X)
  9. Averaging
  10. The framed geodesics $c(X)$
  11. The linking numbers $\iota(X;s)$
  12. The main identity
  13. Unfolding
  14. Proof of Proposition \ref{['prp:ihi']} and \ref{['prp:bmt']}
  15. An integral formula
  16. ...and 13 more sections

Key Result

Theorem 1.1

The function transforms as a Siegel modular form of genus $2$ and level $4$, that is for all $() \in \mathop{\mathrm{Sp}}\nolimits_4(\mathbb{Z})$ with $a\equiv d \equiv () \bmod 4$ and $b\equiv c\equiv () \bmod 4$.

Figures (2)

  • Figure 1: Fundamental domain for $\Gamma'$.
  • Figure 2: The segments of $c'$ which intersect $\ell_{z_1,\infty}$ and $\ell_{z_2,\infty}$ are oriented counter clockwise, hence the intersection numbers are $+1$ and $+2$.

Theorems & Definitions (90)

  • Example 1
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Theorem 1.2
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Remark 3
  • ...and 80 more