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Zeros of theta functions associated with self-dual lattices

Roei Raveh

TL;DR

The paper investigates the zeros of theta functions Θ_{Γ_{4k}} attached to a family of self-dual lattices Γ_{4k}, revealing a dichotomy driven by lattice parity. In the even case (Γ_{8k}), zeros lie on the line Re z = 1/2 within the fundamental domain and are equidistributed along this line with an explicit density derived via the modular lambda function; zeros are simple and heights grow logarithmically with the index. In the odd case (Γ_{8k+4}), zeros avoid the line Re z = 1/2, instead concentrating exponentially near it, with precise counts and decay rates established. The approach reduces the problem to studying zeros of a λ-polynomial p_k through conformal properties of λ, enabling detailed geometric and distributional results along geodesics tied to the level-2 modular curve. These findings illuminate how lattice parity dictates the fine-scale zero geometry of associated theta functions and connect lattice theory with modular forms and hyperbolic geometry.

Abstract

We study the zeros of theta functions $Θ_{Γ_{4k}}$ associated with the lattices $Γ_{4k}$, a family of self-dual lattices generalizing the $\mathsf{E}_{8}$ lattice. Our results show two different behaviors of the zeros according to the lattice parity: When $Γ_{4k}$ is an even lattice, we show that the zeros all lie on the line $\Re z =\frac{1}{2}$ in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line $\Re z = \frac{1}{2}$. However, when the $Γ_{4k}$ is an odd lattice, there are no zeros on the line $\Re z =\frac{1}{2}$, only exponentially close to it. Our argument relies on representing $Θ_{Γ_{4k}}$ as a polynomial in the modular $λ$-function. We then study the zeros of this polynomial and exploit some conformal properties of $λ$ to get our results.

Zeros of theta functions associated with self-dual lattices

TL;DR

The paper investigates the zeros of theta functions Θ_{Γ_{4k}} attached to a family of self-dual lattices Γ_{4k}, revealing a dichotomy driven by lattice parity. In the even case (Γ_{8k}), zeros lie on the line Re z = 1/2 within the fundamental domain and are equidistributed along this line with an explicit density derived via the modular lambda function; zeros are simple and heights grow logarithmically with the index. In the odd case (Γ_{8k+4}), zeros avoid the line Re z = 1/2, instead concentrating exponentially near it, with precise counts and decay rates established. The approach reduces the problem to studying zeros of a λ-polynomial p_k through conformal properties of λ, enabling detailed geometric and distributional results along geodesics tied to the level-2 modular curve. These findings illuminate how lattice parity dictates the fine-scale zero geometry of associated theta functions and connect lattice theory with modular forms and hyperbolic geometry.

Abstract

We study the zeros of theta functions associated with the lattices , a family of self-dual lattices generalizing the lattice. Our results show two different behaviors of the zeros according to the lattice parity: When is an even lattice, we show that the zeros all lie on the line in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line . However, when the is an odd lattice, there are no zeros on the line , only exponentially close to it. Our argument relies on representing as a polynomial in the modular -function. We then study the zeros of this polynomial and exploit some conformal properties of to get our results.

Paper Structure

This paper contains 18 sections, 17 theorems, 108 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Zeros of modular forms
  3. Theta functions and self-dual lattices
  4. Statements and results
  5. Background
  6. Modular forms on $\text{SL}_{2}\left(\mathbb{Z}\right)$
  7. Modular forms on $\Gamma{\left(2\right)}$
  8. Lattices and their associated theta functions
  9. The structure of $\Theta_{\Gamma_{8k}}$ and their $\lambda$-zeros
  10. A couple representations of $\Theta_{\Gamma_{8k}}$
  11. The zeros of the polynomial $p_{k}$
  12. Conformal properties of the modular lambda function
  13. The even case
  14. Proof of Theorem \ref{['thm: Zeros']}
  15. Proof of Theorem \ref{['thm: Zeros Distrbution']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

For all $k\ge1$, the zeros of $\Theta_{\Gamma_{8k}}$ are all simple (except for $\rho$) and lie on the line $\mathcal{L}_{\rho}$. Furthermore, let $\tau_{k,1},\ldots,\tau_{k,\ell}\in \mathcal{L}_{\rho}$ be the zeros of $\Theta_{\Gamma_{8k}}$ ordered with decreasing imaginary value, i.e. $\Im\tau_{k, as $k\to\infty$ and $m=o{\left(k\right)}$.

Figures (10)

  • Figure 1: The fundamental domain $\mathcal{F}$.
  • Figure 2: The zeros of $\Theta_{\Gamma_{4k}}$ where $k\in{\left\{20,35,60\right\}}$.
  • Figure 3: The zeros of $\Theta_{\Gamma_{480}}$ in $\mathcal{F}$.
  • Figure 4: The zeros of $\Theta_{\Gamma_{484}}$ in $\mathcal{F}_{\lambda}$.
  • Figure 5: The fundamental domain $\mathcal{F}_{\lambda}$.
  • ...and 5 more figures

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark
  • Remark
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 27 more