Results 11 to 20 of about 4,688 (128)
Semi-local Units Modulo Cyclotomic Units
Journal of Number Theory, 1999 This paper studies the Galois properties of the quotient of the group of semi-local units by its subgroup of cyclotomic units along the \({\mathbb Z}_p\)-cyclotomic extension. Let \(K\) be an abelian extension of \(\mathbb Q\) and \(G\) its Galois group containing the \(p\)-roots of 1. Let \(\psi\) an irreducible character \(G\). The author defines for
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Cyclotomic polynomials and units in cyclotomic number fields
Journal of Number Theory, 1988 The author proves (theorem 1) that if P(x)\(\neq x\) is a monic irreducible polynomial with integer coefficients such that its resultant with infinitely many cyclotomic polynomials is \(+1\) or -1, then P(x) is a cyclotomic polynomial. From this he deduces a number of interesting corollaries: for example, if \(\alpha\neq 0\) is an algebraic integer ...
Michael Kaminski
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Galois Relations for Cyclotomic Numbers and p-Units
Journal of Number Theory, 1994 Let \(L/\mathbb{Q}\) be an abelian real field of finite degree and let \(f= f_ L\) denote the conductor of \(L\). For \(\zeta_ f= \exp(2\pi i/f)\), put \(\mathbb{Q}(f)= \mathbb{Q}(\zeta_ f)\). Then the cyclotomic number \(\varepsilon_ L\) attached to \(L\) is defined by \(N_{\mathbb{Q} (f)/L} (1- \zeta_ f)\).
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Construction of Bases for the Group of Cyclotomic Units
Journal of Number Theory, 2000 Let \(\varepsilon_n\) be a primitive \(n\)th root of unity and \(D^{(n)}\) the multiplicative group generated by \(1-\varepsilon_n^k\) with \(k\not\equiv 0\bmod n\) modulo roots of unity. The group of cyclotomic units \(C^{(n)}\) is the subgroup of \(D^{(n)}\) consisting of the elements which are units in \(\mathbb{Z}[\varepsilon_n]\).
Marc Conrad
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On the Z_p-ranks of tamely ramified Iwasawa modules
, 2013 For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside
MANABU OZAKI +4 more
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A P‐adic class formula for Anderson t‐modules
Journal of the London Mathematical Society, Volume 113, Issue 4, April 2026.Abstract In 2012, Taelman proved a class formula for L$L$‐series associated to Drinfeld Fq[θ]$\mathbb {F}_q[\theta]$‐modules and considered it as a function field analogue of the Birch and Swinnerton‐Dyer conjecture. Since then, Taelman's class formula has been generalized to the setting of Anderson t$t$‐modules.
Alexis Lucas
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Fermat quotient of cyclotomic units [PDF]
Acta Arithmetica, 1996 Let \(K\) be the cyclotomic field \(\mathbb{Q}(\zeta_{mp})\) of \(mp\)th roots of unity, where \(p\) is an odd prime and \(m>1\) is an odd squarefree number prime to \(p\). Denote by \({\mathcal F}(K)\) the set of all Fermat quotients \(x_u\text{ mod }p\) of the principal units \(u\) of \(K\), defined by the equation \(u=1+ x_up\).
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On the equivariant main conjecture of Iwasawa theory
, 2005 Recently, D. Burns and C. Greither (Invent. Math., 2003) deduced an equivariant version of the main conjecture for abelian number fields. This was the key to their proof of the equivariant Tamagawa number conjecture. A. Huber and G. Kings (Duke Math. J.,
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Ordinary primes for GL2$\operatorname{GL}_2$‐type abelian varieties and weight 2 modular forms
Mathematika, Volume 72, Issue 2, April 2026.Abstract Let A$A$ be a g$g$‐dimensional abelian variety defined over a number field F$F$. It is conjectured that the set of ordinary primes of A$A$ over F$F$ has positive density, and this is known to be true when g=1,2$g=1, 2$, or for certain abelian varieties with extra endomorphisms.
Tian Wang, Pengcheng Zhang
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p$p$‐adic equidistribution and an application to S$S$‐units
Journal of the London Mathematical Society, Volume 113, Issue 3, March 2026.Abstract We prove a Galois equidistribution result for torsion points in Gmn$\mathbb {G}_m^n$ in the p$p$‐adic setting for test functions of the form log|F|p$\log |F|_p$ where F$F$ is a nonzero polynomial with coefficients in the field of complex p$p$‐adic numbers.
Gerold Schefer
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