Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T07:08:32.834Z Has data issue: false hasContentIssue false

2 - Modular stuff

Published online by Cambridge University Press:  19 August 2009

Terry Gannon
Affiliation:
University of Alberta
Get access

Summary

This chapter introduces modular functions and forms, a subject central to the remainder of the book. Some earlier parts of this chapter are beautifully covered in [414].

Section 2.1 supplies the underlying geometry, but can be skimmed on a first reading. In spite of this background material, the theory of modular forms and functions discussed in Sections 2.2 and 2.3 will probably appear as somewhat arbitrary to the uninitiated reader. Section 2.4.1 addresses some of this apparent artificiality, by developing the broader context of automorphic forms.

As explained in the introductory chapter, Moonshine involves unexpected occurrences of modularity. The modularity of Moonshine functions follows from Zhu's Theorem (Theorem 5.3.8). However, the complexity of the underlying mathematics begs the question: Can modularity be established in a more elementary way? The simplest example of Moonshine involves theta functions. Hence we explore the limits and potentials of four classical strategies for proving the modularity of theta functions: Poisson summation, Dirichlet series, the heat kernel and representations of Heisenberg groups (Sections 2.2.3, 2.3.1, 2.3.4 and 2.4.2, respectively).

Moonshine has really only been worked out in genus 1, but conformal field theory tells us that there is an analogue for every genus (Section 6.3.1). It will be much more complicated, but it will be more rewarding because the number theoretic side is much less developed. In otherwords, we will find traces of, for example, the Monster in automorphic forms for the higher mapping class groups Γg,n and Sp2n(ℤ). We include Sections 2.1.4 and 2.3.5 in anticipation of this most natural and significant future development.

Type
Chapter
Information
Moonshine beyond the Monster
The Bridge Connecting Algebra, Modular Forms and Physics
, pp. 104 - 175
Publisher: Cambridge University Press
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Modular stuff
  • Terry Gannon, University of Alberta
  • Book: Moonshine beyond the Monster
  • Online publication: 19 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511535116.003
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Modular stuff
  • Terry Gannon, University of Alberta
  • Book: Moonshine beyond the Monster
  • Online publication: 19 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511535116.003
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Modular stuff
  • Terry Gannon, University of Alberta
  • Book: Moonshine beyond the Monster
  • Online publication: 19 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511535116.003
Available formats
×