Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Measures, modular forms, and summation formulas of Poisson type

Claudia Alfes, Paul Kiefer, Jan Mazáč

TL;DR

The paper establishes a deep link between crystalline, Fourier-eigenmeasure structures and modular forms via a Metaplectic-induced correspondence between $k$-spherical measures and Fourier series. It proves an Mp extsubscript{2}(\,R)-equivariant isomorphism between measures supported on spherical shells and modular-type Fourier data, enabling construction of Fourier eigenmeasures from holomorphic and half-integral weight modular forms. This framework recovers classical Poisson-type summation formulas weighted by modular coefficients and extends naturally to higher dimensions through Hilbert modular forms, highlighting a unifying approach to crystallography, harmonic analysis, and automorphic forms. The results yield explicit summation identities and provide a versatile toolkit for generating and analyzing eigenmeasures with prescribed transformation properties under the Fourier transform.

Abstract

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct $k$-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gonçalves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures.

Measures, modular forms, and summation formulas of Poisson type

TL;DR

The paper establishes a deep link between crystalline, Fourier-eigenmeasure structures and modular forms via a Metaplectic-induced correspondence between -spherical measures and Fourier series. It proves an Mp extsubscript{2}(\,R)-equivariant isomorphism between measures supported on spherical shells and modular-type Fourier data, enabling construction of Fourier eigenmeasures from holomorphic and half-integral weight modular forms. This framework recovers classical Poisson-type summation formulas weighted by modular coefficients and extends naturally to higher dimensions through Hilbert modular forms, highlighting a unifying approach to crystallography, harmonic analysis, and automorphic forms. The results yield explicit summation identities and provide a versatile toolkit for generating and analyzing eigenmeasures with prescribed transformation properties under the Fourier transform.

Abstract

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call -spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct -spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gonçalves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional -spherical measures.
Paper Structure (21 sections, 19 theorems, 97 equations)

This paper contains 21 sections, 19 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Background: Eigenmeasures of the Fourier transform
  3. Background: Jacobi's theta function and Poisson's summation formula
  4. Background: Poisson-type summation formulas and crystallography
  5. Guinand's tempered distribution and the third power of Jacobi's theta function
  6. Main results
  7. Outline
  8. Measures and Fourier series
  9. Schwartz functions and tempered distributions
  10. The metaplectic group and its action on Schwartz functions and Fourier series
  11. Proof of Theorem \ref{['lem:IsoBetweenMeasuresAndFunctions']}
  12. Tempered distributions in odd dimensions
  13. Recovering Poisson type summation formulas
  14. Modular forms for the Fricke group
  15. Poisson type summation formulas involving the coefficients of modular forms
  16. ...and 6 more sections

Key Result

Theorem 1.4

The map is an $\operatorname{Mp}_2(\mathbb{R})$-equivariant isomorphism, i.e., $\theta_{\mu \vert M} = \theta_\mu \vert_{k / 2} M$ for all $M \in \operatorname{Mp}_2(\mathbb{R})$. The inverse is given by

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Example 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 37 more