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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Modular functions arising from some finite groups

Author: Larissa Queen
Journal: Math. Comp. 37 (1981), 547-580
MSC: Primary 20C15; Secondary 10D07, 20D08
MathSciNet review: 628715
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Abstract: In [2] Conway and Norton have assigned a "Thompson series" of the form \[ {q^{ - 1}} + {H_1}q + {H_2}{q^2} + \ldots \] to each element m of the Fischer-Griess "Monster" group M and conjectured that these functions are Hauptmoduls for certain genus-zero modular groups. We have found, for a large number of values of N, all the genus-zero groups between ${\Gamma _0}(N)$ and $PSL(2,R)$ that have Hauptmoduls of the above form, and this provides the necessary verification that the series assigned in [2] to particular elements of M really are such Hauptmoduls. (Atkin and Fong [1] have recently verified that ${H_n}(m)$ really is a character of M for all n.) We compute Thompson series for various finite groups and discuss the differences between these groups and M. We find that the resulting Thompson series are all Hauptmoduls for suitable genus-zero subgroups of $PSL(2,R)$.

References [Enhancements On Off] (What's this?)

    A. O. L. Atkin & P. Fong, A communication at the A. M. S. Conference on Finite Simple Groups, Santa Cruz, 1979.
  • J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI
  • J. H. Conway, R. T. Curtis, S. P. Norton & R. A. Parker, An Atlas of Finite Groups. (In preparation.) L. Queen, Some Relations Between Finite Groups, Lie Groups and Modular Functions, Ph.D. Dissertation, Cambridge, April 1980.
  • J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR 554402, DOI

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Article copyright: © Copyright 1981 American Mathematical Society