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Magnetic orthogonal modular forms

Claudia Alfes, Paul Kiefer

TL;DR

The paper defines magnetic orthogonal modular forms in signature $(2,n)$ and proves that a natural family of meromorphic orthogonal modular forms, $oldsymbol{\omega}_{eta,m}^{ ext{mero}}$, are magnetic by leveraging Borcherds’ additive regularised theta lift. The main mechanism connects weakly holomorphic vector-valued modular forms for the Weil representation to orthogonal modular forms via Borcherds’ lift, yielding explicit divisibility criteria for Fourier coefficients in terms of lattice data and level. A key outcome is a general method to generate magnetic forms from vector-valued inputs with integral Fourier coefficients, valid even when $n$ is odd, with additional linear-combination adjustments when cusp spaces are nontrivial. The results underscore deep links between the arithmetic of Fourier coefficients, Borcherds lifts, and physical theories, including potential interpretations in Calabi–Yau and quantum field contexts.

Abstract

In this note we show that certain meromorphic orthogonal modular forms are magnetic, i.e.\ their Fourier coefficients satisfy special divisibility criteria. These meromorphic orthogonal modular forms are counterparts to the orthogonal cusp forms considered by Oda. We show that the seminal of work of Borcherds implies the magneticity of these forms.

Magnetic orthogonal modular forms

TL;DR

The paper defines magnetic orthogonal modular forms in signature and proves that a natural family of meromorphic orthogonal modular forms, , are magnetic by leveraging Borcherds’ additive regularised theta lift. The main mechanism connects weakly holomorphic vector-valued modular forms for the Weil representation to orthogonal modular forms via Borcherds’ lift, yielding explicit divisibility criteria for Fourier coefficients in terms of lattice data and level. A key outcome is a general method to generate magnetic forms from vector-valued inputs with integral Fourier coefficients, valid even when is odd, with additional linear-combination adjustments when cusp spaces are nontrivial. The results underscore deep links between the arithmetic of Fourier coefficients, Borcherds lifts, and physical theories, including potential interpretations in Calabi–Yau and quantum field contexts.

Abstract

In this note we show that certain meromorphic orthogonal modular forms are magnetic, i.e.\ their Fourier coefficients satisfy special divisibility criteria. These meromorphic orthogonal modular forms are counterparts to the orthogonal cusp forms considered by Oda. We show that the seminal of work of Borcherds implies the magneticity of these forms.
Paper Structure (16 sections, 8 theorems, 49 equations)

This paper contains 16 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Modular forms for the Weil representation
  3. The Weil representation
  4. Modular forms for the Weil representation
  5. Differential operators
  6. Poincaré series
  7. Orthogonal modular forms
  8. The orthogonal upper half plane
  9. Algebraic cycles
  10. Orthogonal modular forms
  11. The functions $\omega$
  12. Theta liftings
  13. A theta function of Borcherds
  14. The Fourier expansion of the additive Borcherds lift
  15. Borcherds' additive regularised theta lift of weakly holomorphic Poincaré series
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that the space of cusp forms of weight $k$ for the Weil representation $\rho_L$ is trivial. Then the functions $\omega_{\beta, m}^{\mathop{\mathrm{mero}}\nolimits}(Z)$ are magnetic.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Remark 2.4
  • Lemma 2.5: Theorem 6.11 iii) in onobook
  • Theorem 4.1
  • ...and 4 more