SPONTANEOUS GENERATION OF MODULAR INVARIANTS

It is possible to compute j(tau) and its modular equations with no per ception of its related classical group structure except at infinity. W e 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 infinity for u = j(tau) and v = j(pr)). We then ask which choice of coefficients of g(u, v) leads to some consist ent Laurent series solution u = u(q) approximate to 1/q, v = u(q(p)) ( where q = exp 2 pi i tau). It is conjectured that if the same Lament s eries 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 t he set of ''replicable functions,'' which include more classical modul ar 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. C omputationally, this process stands alone, and, in a sense, modular in variants arise ''spontaneously.''