Spontaneous Generation of Modular Invariants

Authors:
Harvey Cohn and John McKay

Journal:
Math. Comp. **65** (1996), 1295-1309

MSC (1991):
Primary 11F11, 20D08

DOI:
https://doi.org/10.1090/S0025-5718-96-00726-0

MathSciNet review:
1344608

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Abstract | References | Similar Articles | Additional Information

Abstract: It is possible to compute and its modular equations with no perception of its related classical group structure except at . We start by taking, for prime, an unknown ``-Newtonian'' polynomial equation with arbitrary coefficients (based only on Newton's polygon requirements at for and ). We then ask which choice of coefficients of leads to some consistent Laurent series solution , (where . It is conjectured that if the same Laurent series works for -Newtonian polynomials of two or more primes , 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 . A demonstration for orders and is done by computation. More remarkably, if the same series works for the -Newtonian polygons of 15 special ``Fricke-Monster'' values of , then is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise ``spontaneously.''

**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*, 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*, 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 -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

Email:
hihcc@cunyvm.edu

**John McKay**

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

Email:
mckay@vax2.concordia.ca

DOI:
https://doi.org/10.1090/S0025-5718-96-00726-0

Keywords:
Modular functions,
modular equations,
replicable functions

Received by editor(s):
January 13, 1995

Article copyright:
© Copyright 1996
American Mathematical Society