Spontaneous generation of modular invariants

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.”