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Locally and Polar Harmonic Maass Forms for Orthogonal Groups of Signature $(2, n)$

Paul Kiefer

Abstract

We generalize the notions of locally and polar harmonic Maass forms to general orthogonal groups of signature $(2, n)$ with singularities along real analytic and algebraic cycles. We prove a current equation for locally harmonic Maass forms and recover the Fourier expansion of the Oda lift involving cycle integrals. Moreover, using the newly defined polar harmonic Maass forms, we prove that meromorphic modular forms with singularities along special divisors are orthogonal to cusp forms with respect to a regularized Petersson inner product. Using this machinery, we derive a duality theorem involving cycle integrals of meromorphic modular forms along real analytic cycles and cycle integrals of locally harmonic Maass forms along algebraic cycles.

Locally and Polar Harmonic Maass Forms for Orthogonal Groups of Signature $(2, n)$

Abstract

We generalize the notions of locally and polar harmonic Maass forms to general orthogonal groups of signature with singularities along real analytic and algebraic cycles. We prove a current equation for locally harmonic Maass forms and recover the Fourier expansion of the Oda lift involving cycle integrals. Moreover, using the newly defined polar harmonic Maass forms, we prove that meromorphic modular forms with singularities along special divisors are orthogonal to cusp forms with respect to a regularized Petersson inner product. Using this machinery, we derive a duality theorem involving cycle integrals of meromorphic modular forms along real analytic cycles and cycle integrals of locally harmonic Maass forms along algebraic cycles.

Paper Structure

This paper contains 19 sections, 22 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Outlook
  3. Outline
  4. Orthogonal Modular Forms
  5. Orthogonal Upper Half Plane
  6. Real Analytic and Algebraic Cycles
  7. Orthogonal Modular Forms
  8. The $\xi$-Operator on the Orthogonal Upper Half Plane
  9. Hermitian Geometry on the Orthogonal Upper Half Plane
  10. Some Formulas Involving the $\xi$-Operator
  11. Locally Harmonic Maass Forms
  12. Locally Harmonic Maass Forms
  13. Integrals Around Real Analytic Cycles
  14. A Current Equation For Locally Harmonic Maass Forms
  15. The Fourier Expansion of the Oda Lift
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exists a real analytic $(n,n-1)$-form with values in $\mathcal{L}_{-\kappa}$ denoted by $\Omega_{\mu}^{\operatorname{cusp}}$ that is harmonic outside a real analytic cycle $C_\mu$ such that $\xi_{-\kappa} \Omega_\mu^{\operatorname{cusp}} = \omega_\mu^{\operatorname{cusp}}$. The space generated

Theorems & Definitions (48)

  • Theorem 1.1: see \ref{['thm:LocallyHarmonicMF']}
  • Theorem 1.2: see \ref{['thm:CurrentEq']}
  • Theorem 1.3: see \ref{['thm:PolarHarmonicMF']}
  • Theorem 1.4: see \ref{['thm:OrthogonalityOnCF']}
  • Theorem 1.5: see \ref{['thm:DualityTheorem']}
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • ...and 38 more