Reference : Splitting fields, prime decomposition and modular forms]{Splitting fields of X^n-X-1...
 Document type : E-prints/Working papers : Already available on another site Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/51645
 Title : Splitting fields, prime decomposition and modular forms]{Splitting fields of X^n-X-1 (particularly for n=5), prime decomposition and modular forms Language : English 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) >] Publication date : 2022 Peer reviewed : No Abstract : [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 $\mathbb Q(\sqrt{19 \cdot 151})$. Permalink : http://hdl.handle.net/10993/51645 source URL : https://arxiv.org/abs/2206.08116

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