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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.”

- D. Alexander, C. Cummins, J. McKay, and C. Simons,
*Completely replicable functions*, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 87–98. MR**1200252**, DOI https://doi.org/10.1017/CBO9780511629259.010 - A. O. L. Atkin and J. Lehner,
*Hecke operators on $\Gamma _{0}(m)$*, Math. Ann.**185**(1970), 134–160. MR**268123**, DOI https://doi.org/10.1007/BF01359701 - Harvey Cohn,
*Fricke’s two-valued modular equations*, Math. Comp.**51**(1988), no. 184, 787–807. MR**935079**, DOI https://doi.org/10.1090/S0025-5718-1988-0935079-4 - Harvey Cohn,
*A numerical survey of the reduction of modular curve genus by Fricke’s involutions*, Number theory (New York, 1989/1990) Springer, New York, 1991, pp. 85–104. MR**1124636** - Harvey Cohn,
*How branching properties determine modular equations*, Math. Comp.**61**(1993), no. 203, 155–170. MR**1195433**, DOI https://doi.org/10.1090/S0025-5718-1993-1195433-7 - H. Cohn,
*Half-step modular equations*, Math. of Comput.**64**(1995), 1267–1285. - J. H. Conway and S. P. Norton,
*Monstrous moonshine*, Bull. London Math. Soc.**11**(1979), no. 3, 308–339. MR**554399**, DOI https://doi.org/10.1112/blms/11.3.308 - David Ford, John McKay, and Simon Norton,
*More on replicable functions*, Comm. Algebra**22**(1994), no. 13, 5175–5193. MR**1291027**, DOI https://doi.org/10.1080/00927879408825127 - R. Fricke,
*Lehrbuch der Algebra III (Algebraische Zahlen)*, Vieweg, Braunschweig, 1928. - R. Fricke,
*Über die Berechnung der Klasseninvarianten*, Acta Arith.**52**(1929), 257–279. - D.H. Lehmer,
*Properties of coefficients of the modular invariant $J(\tau )$*, Amer. J. Math.**64**(1942), 488–502. - K. Mahler,
*On a class of non-linear functional equations connected with modular functions*, J. Austral. Math. Soc. Ser. A**22**(1976), no. 1, 65–118. MR**441867**, DOI https://doi.org/10.1017/s1446788700013367 - Yves Martin,
*On modular invariance of completely replicable functions*, (preprint). - John McKay and Hubertus Strauss,
*The $q$-series of monstrous moonshine and the decomposition of the head characters*, Comm. Algebra**18**(1990), no. 1, 253–278. MR**1037906**, DOI https://doi.org/10.1080/00927879008823911

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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

Keywords:
Modular functions,
modular equations,
replicable functions

Received by editor(s):
January 13, 1995

Article copyright:
© Copyright 1996
American Mathematical Society