On some properties of polynomials related to hypergeometric modular forms (Q874901)

The authors study certain polynomials \(P^{(r)}_m (x)\) with rational coefficients indexed by positive integers \(r,m\) with \(0\leq r \leq 5\). These polynomials are defined using hypergeometric series and have the property that the roots of their reductions mod \(p\) for certain specific primes \(p\) are related to the supersingular \(j\)-invariants in characteristic \(p\). The authors compute explicitly the discriminant of these polynomials and prove their irreducibility in certain cases. Finally they verify that the Galois group of \(P^{(r)}_m (x)\) over the rationals is the full symmetric group for all \(r\) and for \(m<150\).