Spontaneous Generation of Modular Invariants |
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Authors: | Harvey Cohn John McKay |
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Institution: | Department of Mathematics, City College (Cuny), New York, New York 10031 ; Department of Computer Science, Concordia University, Montreal, Quebec, Canada H3G 1M8 |
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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.' |
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Keywords: | Modular functions modular equations replicable functions |
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