## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# 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
Preview not available

## 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."

• 1295
• 1296
• 1297
• 1298
• 1299
• 1300
• 1301
• 1302
• 1303
• 1304
• 1305
• 1306
• 1307
• 1308
• 1309