Arithmetic of Certain Hypergeometric Modular Forms

In a recent paper, Kaneko and Zagier studied a sequence of modular forms Fk(z) which are solutions of a certain second order differential equation. They studied the polynomials e Fk(j) = Y τ∈H/Γ−{i,ω} (j − j(τ))τ k, where ω = e2πi/3 and H/Γ is the usual fundamental domain of the action of SL2(Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e Fp−1(j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.