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

Chandrashekhar B. Khare; Alfio Fabio La Rosa; Gabor Wiese

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

Number Theory 2022-11-01 v3

Authors: Chandrashekhar B. Khare , Alfio Fabio La Rosa , Gabor Wiese

Abstract

We study the splitting fields of the family of polynomials $f_n(X)= X^n-X-1$ . 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 $n \leq 4$ 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 $N_p(f_5)$ to mod $5$ modular forms over $\mathbb Q$ , and to characteristic 0, parallel weight 1 Hilbert modular forms over $\mathbb Q(\sqrt{19 \cdot 151})$ .

Keywords

splitting fields polynomials X^n-X-1 prime decomposition modular forms hilbert modular forms number theory

Cite

@article{arxiv.2206.08116,
  title  = {Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms},
  author = {Chandrashekhar B. Khare and Alfio Fabio La Rosa and Gabor Wiese},
  journal= {arXiv preprint arXiv:2206.08116},
  year   = {2022}
}

Comments

14 pages, v2: much smaller polynomial thanks to J\"urgen Kl\"uners; v3: extended exposition of liftings of projective representations

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

Abstract

Keywords

Cite

Comments

Related papers