"Moonshine" of Finite GroupsEuropean Mathematical Society, 2010 - 76 ページ This is an almost verbatim reproduction of the author's lecture notes written in 1983-84 at Ohio State University, Columbus. A substantial update is given in the bibliography. Over the last 20 plus years there has been energetic activity in the field of finite simple group theory related to the monster simple group. Most notably, influential works have been produced in the theory of vertex operator algebras from research that was stimulated by the moonshine of the finite groups. Still, we can ask the same questions now that we did 30-40 years ago: What is the monster simple group? Is it really related to the theory of the universe as it was vaguely so envisioned? What lies behind the moonshine phenomena of the monster group? It may appear that we have only scratched the surface. These notes are primarily reproduced for the benefit of readers who wish to start learning about modular functions used in moonshine. |
目次
Preface | 1 |
Moonshine of finite groups | 31 |
Multiplicative product of n functions | 54 |
Appendix Genus zero discrete groups | 65 |
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a₁ Amer assume Atkin-Lehner involution bd(c² characters of G coefficients compact Riemann surface completes the proof complex number compute conjugate converges curve cusp form cusps of To(N Dedekind Dedekind eta function Define Definition Differential discrete subgroup equivalent Exercise f(rz finite group form of weight Fuchsian group function field function of q fundamental domain genus Hecke character Hence holds Im(z implies inequivalent cusps invariant under To(N k(Rr Koike lecture notes Lemma Let G Let h lim f'(x limx Math meromorphic function modular forms modular function Monster moonshine natural number notation number field partition Phys primitive cusp form quadratic character R. E. Borcherds r₁ Re(z satisfies the condition Show Suppose T₁ Theorem V(xi vertex operator algebras δε ηπ πί πίζ