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Spontaneous Generation of Modular Invariants

Harvey Cohn and John Mckay
Mathematics of Computation
Vol. 65, No. 215 (Jul., 1996), pp. 1295-1309
Stable URL: http://www.jstor.org/stable/2153808
Page Count: 15
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Spontaneous Generation of Modular Invariants
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Abstract

It is possible to compute j(τ) and its modular equations with no perception of its related classical group structure except at ∞. We start by taking, for p prime, an unknown "p-Newtonian" polynomial equation g(u, ν) = 0 with arbitrary coefficients (based only on Newton's polygon requirements at ∞ for u = j(τ) and v = j(pτ)). We then ask which choice of coefficients of g(u, ν) leads to some consistent Laurent series solution u = u(q) ≈ 1/q, ν = u(qp) (where q = exp 2π iτ). 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(τ). 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(τ) is (essentially) determined uniquely. Computationally, this process stands alone, and, in a sense, modular invariants arise "spontaneously."

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