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Spontaneous Generation of Modular Invariants

Authors: Harvey Cohn and John McKay
Journal: Math. Comp. 65 (1996), 1295-1309
MSC (1991): Primary 11F11, 20D08
MathSciNet review: 1344608
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Abstract: It is possible to compute $j(\tau )$ and its modular equations with no perception of its related classical group structure except at $\infty $. We start by taking, for $p$ prime, an unknown ``$p$-Newtonian'' polynomial equation $g(u,v)=0$ with arbitrary coefficients (based only on Newton's polygon requirements at $\infty $ for $u=j(\tau )$ and $v=j(p\tau )$). We then ask which choice of coefficients of $g(u,v)$ leads to some consistent Laurent series solution $u=u(q)\approx 1/q$, $v=u(q^{p})$ (where $q=\exp 2\pi i\tau )$. It is conjectured that if the same Laurent series $u(q)$ works for $p$-Newtonian polynomials of two or more primes $p$, then there is only a bounded number of choices for the Laurent series (to within an additive constant). These choices are essentially from the set of ``replicable functions,'' which include more classical modular invariants, particularly $u=j(\tau )$. A demonstration for orders $p=2$ and $3$ is done by computation. More remarkably, if the same series $u(q)$ works for the $p$-Newtonian polygons of 15 special ``Fricke-Monster'' values of $p$, then $(u=)j(\tau )$ is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.''

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

  • 1. D. Alexander, C. Cummins, J. McKay, and C.Simons, Completely replicable functions, Groups, Combinatorics and Geometry (M.W. Liebeck and J. Saxl, eds.) (1992), 87--98; (LMS Lecture Note Series, vol. 165), Cambridge Univ. Press . MR 94g:11029
  • 2. A.O.L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134--160. MR 42:3022
  • 3. H. Cohn, Fricke's two-valued modular equations, Math. of Comput. 51 (1988), 787--807. MR 89f:11064
  • 4. H. Cohn, A numerical survey of the reduction of modular curve genus by Fricke's involutions, Springer Verlag, 1991, pp. 85--104; Number Theory, New York Seminar (1989-90) . MR 92f:11060
  • 5. H. Cohn, How branching properties determine modular equations, Math. of Comput. 61 (1993), 155--170. MR 93k:11036
  • 6. H. Cohn, Half-step modular equations, Math. of Comput. 64 (1995), 1267--1285. MR 96a:11038
  • 7. J.H. Conway and S.P. Norton, Monstrous moonshine, Bull. Lond. Math. Soc. 11 (1979), 308--339. MR 81j:20028
  • 8. D. Ford, J. McKay and S. Norton, More on replicable functions, Comm. in Algebra 13 (1994), 5175--5193. MR 95i:11036
  • 9. R. Fricke, Lehrbuch der Algebra III (Algebraische Zahlen), Vieweg, Braunschweig, 1928.
  • 10. R. Fricke, Über die Berechnung der Klasseninvarianten, Acta Arith. 52 (1929), 257--279.
  • 11. D.H. Lehmer, Properties of coefficients of the modular invariant $J(\tau )$, Amer. J. Math. 64 (1942), 488--502. MR 3:272c
  • 12. K. Mahler, On a class of non-linear functional equations connected with modular equations, J. Austral. Math. Soc. 22A (1976), 65--118. MR 56:258
  • 13. Yves Martin, On modular invariance of completely replicable functions, (preprint).
  • 14. J. McKay and H. Strauss, The $q$-series decompositions of monstrous moonshine and the decomposition of the head characters, Comm. in Algebra 18 (1990), 253--278. MR 90m:11065

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Additional Information

Harvey Cohn
Affiliation: Department of Mathematics, City College (Cuny), New York, New York 10031
Address at time of publication: IDA, Bowie, Maryland 20715-4300

John McKay
Affiliation: Department of Computer Science, Concordia University, Montreal, Quebec, Canada H3G 1M8

Keywords: Modular functions, modular equations, replicable functions
Received by editor(s): January 13, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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