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   Splitting fields of X^n-X-1 (particularly for n=5), prime decomposition and modular forms; La Rosa, Alfio Fabio  ; Wiese, Gabor in Expositiones Mathematica (in press) 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 ... [more ▼] 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 $\mathbb Q(\sqrt{19 \cdot 151})$. [less ▲] Detailed reference viewed: 74 (3 UL) Revisiting Poincaré's Theorem on presentations of discontinuous groups via fundamental polyhedra; Kiefer, Ann ; in Expositiones Mathematica (2015), 33(4), 401--430 We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discon- tinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the ... [more ▼] We give a new self-contained proof of Poincaré's Polyhedron Theorem on presentations of discon- tinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the theory of covering spaces, but only makes use of basic geometric concepts. In a sense one hence obtains a proof that is of a more constructive nature than most known proofs. [less ▲] Detailed reference viewed: 48 (2 UL)Harmonic Analysis of Weighted LP-Algebras; Molitor-Braun, Carine in Expositiones Mathematica (2012), 30(2), 124-153 Detailed reference viewed: 85 (19 UL) |
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