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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.''

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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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