Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms
| Autor: | Khare, Chandrashekhar B., La Rosa, Alfio Fabio, Wiese, Gabor |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathbb Q(\sqrt{19 \cdot 151})$. Comment: 14 pages, v2: much smaller polynomial thanks to J\"urgen Kl\"uners; v3: extended exposition of liftings of projective representations |
| Databáze: | arXiv |
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