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

2022/06/16 by Khare, Chandrashekhar B., La Rosa, Alfio Fabio, Wiese, Gabor
#11F11 #11F33 #11F41 #11F80 (primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.08116

Abstract

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

Related

• Splitting primes
• Splitting of primes in number fields generated by points on some modular curves
• Explicit factorization of xⁿ-1∈ \mathbb F_q[x]
• Hilbert's Irreducibility, Modular Forms, and Computation of Certain Galois Groups
• Splitting of integer polynomials over fields of prime order