Results 21 to 30 of about 130 (120)
Cyclotomic extensions of number fields
Indagationes Mathematicae, 2003 Let \(K\) be a number field, \(l\) a prime number, and \(\zeta_l\) a primitive \(l\)th root of unity. The paper is devoted to the cyclotomic extension \(K(\zeta_l)/K\), giving explicit formulae for the discriminant, conductor, and different of this extension.
Cohen, Henri +2 more
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Lower bounds for heights in cyclotomic extensions
Journal of Number Theory, 2010 Let \(f(x) = a_{0} \prod_{i=1}^{d} (x-\alpha_{i}) = a_{0}x^{n}+a_{1}x^{n-1}+ \cdots + a_{n}\) be an irreducible polynomial with integer coefficients. The Mahler measure of \(f\) is defined to be \(M(f) =\prod_{i=1}^{d}\max \{ 1, |\alpha_{i}| \}\). If \(\alpha \neq 0\) is a root of \(f\), then the absolute logarithmic height \(h(\alpha)\) of \(\alpha ...
Ishak, M.I.M. +3 more
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On 7‐adic Galois representations for elliptic curves over Q$\mathbb {Q}$
Proceedings of the London Mathematical Society, Volume 133, Issue 2, August 2026.Abstract In recent years, significant progress has been made on Mazur's Program B, with many authors beginning a systematic classification of all possible images of p$p$‐adic Galois representations attached to elliptic curves over Q$\mathbb {Q}$. Currently, the classification is only complete for p∈{2,3,13,17}$p \in \lbrace 2,3,13,17\rbrace$.
Lorenzo Furio, Davide Lombardo
wiley +1 more source
Renormalization techniques for inflation systems and some of their applications
Acta Crystallographica Section A, Volume 82, Issue 4, Page 294-304, July 2026.In this work, renormalization methods for quantities related to the diffraction of inflation systems are surveyed.Exact renormalization techniques are important and powerful, particularly for inflation‐generated systems. We review recent results in this direction.
Michael Baake +4 more
wiley +1 more source
Sylow subgroups and the number of irreducible characters of degrees divisible by a prime p$p$
Bulletin of the London Mathematical Society, Volume 58, Issue 7, July 2026.Abstract Let G$G$ be a finite group and p$p$ be a prime. We establish an upper bound for the derived length of a Sylow p$p$‐subgroup of G$G$ in terms of the number of irreducible characters of G$G$ whose degrees are divisible by p$p$. We also prove that if B$B$ is a p$p$‐block of a finite p$p$‐solvable group G$G$ with defect group D$D$, then the ...
James P. Cossey +3 more
wiley +1 more source
Abelian threefolds with imaginary multiplication
Journal of the London Mathematical Society, Volume 114, Issue 1, July 2026.Abstract Let A$A$ be an abelian threefold defined over a number field K$K$ with potential multiplication by an imaginary quadratic field M$M$. Under mild assumptions on K$K$, if A$A$ has signature (2,1) and the multiplication by M$M$ is defined over KM$KM$, we attach to A$A$ an elliptic curve defined over K$K$ with potential complex multiplication by M$
Francesc Fité, Pip Goodman
wiley +1 more source
The degrees of the cyclotomic extension fields
Linear Algebra and its Applications, 1993 The following question is investigated: ``Suppose \(r\) and \(s\) are relatively prime positive integers and \(\xi_ r\), \(\xi_ s\), \(\xi_{rs}\) primitive roots of unity. When for positive integers \(a\), \(b\) and \(c\) is there a field \(K\) of characteristic zero with \(| K(\xi_{rs}): K|=a\), \(| K(\xi_ r): K| =b\), and \(| K(\xi_ s):K| =c ...
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Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions [PDF]
, 2010 This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of $\mathbb{F}_{q^6}^{\times}$, for $q \equiv 1 \bmod{6}$. Our result arises from considering the Weil restriction of scalars of this group from $\mathbb{F}_{q^6}$ to $\mathbb{F}_{q^2}$, and provides efficiency improvements for both pairing-based and ...
Robert Granger, Michael Scott
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When is a 2-Power Cyclotomic Extension cyclic?
, 2023 This paper characterizes the cyclicity property of $2$-power cyclotomic extensions through various means: the structure of the Galois groups, the nature of their subextensions, tower decompositions, and, most importantly, specific conditions on the base field.
Marques, Sophie, Mrema, Elizabeth
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On the Lang–Trotter conjecture for Siegel modular forms
Mathematika, Volume 72, Issue 3, July 2026.Abstract Let f$f$ be a genus‐two cuspidal Siegel eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated with f$f$, generalizing the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues ap$a_p$ of f$f$, and obtain upper
Arvind Kumar, Moni Kumari, Ariel Weiss
wiley +1 more source