Splitting fields of [formula omitted] (particularly for [formula omitted]), prime decomposition and modular forms

Abstract

We study the splitting fields of the family of polynomials fn(X)=Xn−X−1. This family of polynomials has been much studied in the literature and has some remarkable properties. In Serre (2003), Serre related the function on primes Np(fn), for a fixed n≤4 and p a varying prime, which counts the number of roots of fn(X) in Fp to coefficients of modular forms. We study the case n=5, and relate Np(f5) to mod 5 modular forms over Q, and to characteristic 0, parallel weight 1 Hilbert modular forms over Q(19⋅151).