On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials

Abstract

Let p be a prime number. The supersingular polynomial S p (x) in characteristic p is the polynomial over F p whose roots in F p are the supersingular j-invariants of elliptic curves over F p . In this paper we study the Newton polygons of certain rational lifts of S p (x) connected to the theory of Jacobi polynomials. As a corollary to our results on the Newton polygons, we obtain new cases of irreducibility for these supersingular lifts, providing more evidence for the general irreducibility conjecture of Mahlburg and Ono.