Splitting fields of Xn−X−1 (particularly for n=5), prime decomposition and modular forms

Abstract

We study the splitting fields of the family of polynomials $f_n\left(X\right)=X^n-X-1$. 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 $N_p\left(f_n\right)$, for a fixed $n\le 4$ and $p$ a varying prime, which counts the number of roots of $f_n\left(X\right)$ in $F_p$ to coefficients of modular forms. We study the case $n=5$, and relate $N_p\left(f_5\right)$ to mod 5 modular forms over $Q$, and to characteristic 0, parallel weight 1 Hilbert modular forms over $Q\left(\sqrt{19\cdot 151}\right)$.