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Search: MSC category 11R18 ( Cyclotomic extensions )

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1. CMB 2016 (vol 59 pp. 624)

Otsubo, Noriyuki
Homology of the Fermat Tower and Universal Measures for Jacobi Sums
We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a similar manner to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Keywords:Fermat curves, Ihara-Anderson theory, Jacobi sums
Categories:11S80, 11G15, 11R18

2. CMB 2015 (vol 58 pp. 423)

Yamagishi, Masakazu
Resultants of Chebyshev Polynomials: The First, Second, Third, and Fourth Kinds
We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

Keywords:resultant, Chebyshev polynomial, cyclotomic polynomial
Categories:11R09, 11R18, 12E10, 33C45

3. CMB 2011 (vol 54 pp. 645)

Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
An Extension of Craig's Family of Lattices
Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields
Categories:11H31, 11H55, 11H50, 11R18, 11R04

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