[en] 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 $\mathbb Q(\sqrt{19 \cdot 151})$.
Disciplines :
Mathematics
DOI :
10.1016/j.exmath.2023.02.007
Author, co-author :
Khare, Chandrashekhar
La Rosa, Alfio Fabio ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Wiese, Gabor ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
External co-authors :
yes
Language :
English
Title :
Splitting fields of X^n-X-1 (particularly for n=5), prime decomposition and modular forms
Publication date :
2023
Journal title :
Expositiones Mathematica
ISSN :
0723-0869
Publisher :
Elsevier, Mannheim, Netherlands
Special issue title :
Special Issue in Honor of B. Edixhoven (1962-2022)