Reducible modular differential equations, Jacobi pole divisors, and supersingular lifts

Abstract

We study the reducible parameter values of a second-order modular differential equation for the full modular group and show that their pole data is controlled by a single polynomial linking three a priori different structures. The residue conditions for the associated weight two form are exactly the Stieltjes equilibrium equations for four shifted Jacobi families. This determines the poles uniquely and produces a canonical divisor on $X(1)$ supported on the elliptic arc. After the standard eta-normalization, with $\ell=6r$, the differential equation takes the Kaneko-Zagier form, and the same Jacobi polynomial is precisely the polynomial factor in its distinguished modular solution. When $\ell$ is prime, this polynomial is an $\ell$-integral characteristic-zero lift of the non-elliptic supersingular polynomial. Consequently, its splitting field over $\Q_\ell$ is unramified of degree at most two, with Frobenius cycle structure determined by the supersingular $j$-invariants. Complete splitting occurs exactly when every supersingular $j$-invariant in characteristic $\ell$ is defined over $\F_\ell$, equivalently when the Fricke quotient $X_0^+(\ell)$ has genus zero. These are Ogg's primes, which are exactly the prime divisors of the order of the Monster. A common singular Sturm-Liouville equation further gives strict interlacing for consecutive reducible parameters. Finally, we prove that the canonical representatives of the poles are transcendental although their $j$-invariants are algebraic, and determine their limiting distribution and endpoint scales on the modular arc.