[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 :
Mathématiques
Auteur, co-auteur :
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)
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Splitting fields of X^n-X-1 (particularly for n=5), prime decomposition and modular forms
Date de publication/diffusion :
2023
Titre du périodique :
Expositiones Mathematica
ISSN :
0723-0869
Maison d'édition :
Elsevier, Mannheim, Pays-Bas
Titre particulier du numéro :
Special Issue in Honor of B. Edixhoven (1962-2022)
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