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Results 126 - 150 of 197 |
126. CMB 2003 (vol 46 pp. 473)
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A Multiplicative Analogue of Schur's Tauberian Theorem A theorem concerning the asymptotic behaviour of partial sums of the
coefficients of products of Dirichlet series is proved using properties of
regularly varying functions. This theorem is a multiplicative analogue of
Schur's Tauberian theorem for power series.
Category:11N45 |
127. CMB 2003 (vol 46 pp. 229)
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Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials Recently there has been tremendous interest in counting the number of
integral points in $n$-dimen\-sional tetrahedra with non-integral
vertices due to its applications in primality testing and factoring
in number theory and in singularities theory. The purpose of this
note is to formulate a conjecture on sharp upper estimate of the
number of integral points in $n$-dimensional tetrahedra with
non-integral vertices. We show that this conjecture is true for
low dimensional cases as well as in the case of homogeneous
$n$-dimensional tetrahedra. We also show that the Bernoulli
polynomials play a role in this counting.
Categories:11B75, 11H06, 11P21, 11Y99 |
128. CMB 2003 (vol 46 pp. 178)
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Sur les invariants d'Iwasawa des tours cyclotomiques We carry out the computation of the Iwasawa invariants $\rho^T_S$,
$\mu^T_S$, $\lambda^T_S$ associated to abelian $T$-ramified
over the finite steps $K_n$ of the cyclotomic
$\mathbb{Z}_\ell$-extension $K_\infty/K$ of a number field of
$\CM$-type.
Nous d\'eterminons explicitement les param\'etres d'Iwasawa
$\rho^T_S$, $\mu^T_S$, $\lambda^T_S$ des $\ell$-groupes de
$S$-classes $T$-infinit\'esimales $\Cl^T_S (K_n)$ attach\'es aux
\'etages finis de la $\mathbb{Z}_\ell$-extension cyclotomique
$K_\infty/K$ d'un corps de nombres \`a conjugaison complexe.
Categories:11R23, 11R37 |
129. CMB 2003 (vol 46 pp. 157)
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Torsion Points on Certain Families of Elliptic Curves Fix an elliptic curve $y^2 = x^3+Ax+B$, satisfying $A,B \in \ZZ$,
$A\geq |B| > 0$. We prove that the $\QQ$-torsion subgroup is one of
$(0)$, $\ZZ/3\ZZ$, $\ZZ/9\ZZ$. Related numerical calculations are
discussed.
Category:11G05 |
130. CMB 2003 (vol 46 pp. 26)
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Remarques sur les points rationnels des variétés de Fermat Soit $K$ un corps de nombres de degr\'e sur $\mathbb{Q}$ inf\'erieur
ou \'egal \`a $2$. On se propose dans ce travail de faire quelques
remarques sur la question de l'existence de deux \'el\'ements non nuls
$a$ et $b$ de $K$, et d'un entier $n\geq 4$, tels que l'\'equation
$ax^n + by^n = 1$ poss\`ede au moins trois points distincts non
triviaux. Cette \'etude se ram\`ene \`a la recherche de points
rationnels sur $K$ d'une vari\'et\'e projective dans $\mathbb{P}^5$ de
dimension $3$, ou d'une surface de $\mathbb{P}^3$.
Category:11D41 |
131. CMB 2003 (vol 46 pp. 39)
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Power Residue Criteria for Quadratic Units and the Negative Pell Equation Let $d>1$ be a square-free integer. Power residue criteria for the
fundamental unit $\varepsilon_d$ of the real quadratic fields $\QQ
(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by
means of class field theory. These results will then be interpreted
as criteria for the solvability of the negative Pell equation $x^2 -
dp^2 y^2 = -1$. The most important solvability criterion deals with
all $d$ for which $\QQ (\sqrt{-d})$ has an elementary abelian 2-class
group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.
Categories:11R11, 11R27 |
132. CMB 2003 (vol 46 pp. 71)
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The Number of Fields Generated by the Square Root of Values of a Given Polynomial The $abc$-conjecture is applied to various questions involving the
number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary
over integers $n$.
Categories:11N32, 11D41 |
133. CMB 2003 (vol 46 pp. 149)
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The Ramification Polygon for Curves over a Finite Field A Newton polygon is introduced for a ramified point of a Galois
covering of curves over a finite field. It is shown to be determined
by the sequence of higher ramification groups of the point. It gives
a blowing up of the wildly ramified part which separates the branches
of the curve. There is also a connection with local reciprocity.
Category:11G20 |
134. CMB 2002 (vol 45 pp. 606)
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Postcards from the Edge, or Snapshots of the Theory of Generalised Moonshine We begin by reviewing Monstrous Moonshine. The impact of Moonshine on
algebra has been profound, but so far it has had little to teach
number theory. We introduce (using `postcards') a much larger context
in which Monstrous Moonshine naturally sits. This context suggests
Moonshine should indeed have consequences for number theory. We
provide some humble examples of this: new generalisations of Gauss
sums and quadratic reciprocity.
Categories:11F22, 17B67, 81T40 |
135. CMB 2002 (vol 45 pp. 466)
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A Note on the Automorphic Langlands Group Langlands has conjectured the existence of a universal group, an
extension of the absolute Galois group, which would play a fundamental
role in the classification of automorphic representations. We shall
describe a possible candidate for this group. We shall also describe
a possible candidate for the complexification of Grothendieck's
motivic Galois group.
Categories:11R39, 22E55 |
136. CMB 2002 (vol 45 pp. 364)
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Mellin Transforms of Whittaker Functions In this note we show that for an arbitrary reductive Lie group
and any admissible irreducible Banach representation the Mellin
transforms of Whittaker functions extend to meromorphic functions.
We locate the possible poles and show that they always lie along
translates of walls of Weyl chambers.
Categories:11F30, 22E30, 11F70, 22E45 |
137. CMB 2002 (vol 45 pp. 337)
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Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, non-split Cartan Categories:11G05, 11F80 |
138. CMB 2002 (vol 45 pp. 428)
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Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 |
139. CMB 2002 (vol 45 pp. 196)
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Mahler Measures Close to an Integer We prove that the Mahler measure of an algebraic number cannot be too
close to an integer, unless we have equality. The examples of certain
Pisot numbers show that the respective inequality is sharp up to a
constant. All cases when the measure is equal to the integer are
described in terms of the minimal polynomials.
Keywords:Mahler measure, PV numbers, Salem numbers Categories:11R04, 11R06, 11R09, 11J68 |
140. CMB 2002 (vol 45 pp. 257)
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Modular Forms Associated to Theta Functions We use the theory of Jacobi-like forms to construct modular forms for a
congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear
combinations of products of certain theta functions.
Categories:11F11, 11F27, 33D10 |
141. CMB 2002 (vol 45 pp. 247)
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On a Few Diophantine Equations Related to Fermat's Last Theorem We combine the deep methods of Frey, Ribet, Serre and Wiles with some
results of Darmon, Merel and Poonen to solve certain explicit
diophantine equations. In particular, we prove that the area of a
primitive Pythagorean triangle is never a perfect power, and that each
of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no
non-trivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.
Keywords:Diophantine equations Category:11D41 |
142. CMB 2002 (vol 45 pp. 231)
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Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440--451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |
143. CMB 2002 (vol 45 pp. 168)
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Biquadratic Extensions with One Break We explicitly describe, in terms of indecomposable $\mathbb{Z}_2
[G]$-modules, the Galois module structure of ideals in totally
ramified biquadratic extensions of local number fields with only
one break in their ramification filtration. This paper completes
work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].
Categories:11S15, 20C11 |
144. CMB 2002 (vol 45 pp. 220)
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Globalization of Distinguished Supercuspidal Representations of $\GL(n)$ An irreducible supercuspidal representation $\pi$ of $G=
\GL(n,F)$, where $F$ is a nonarchimedean local field of
characteristic zero, is said to be ``distinguished'' by a
subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if
$\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue
of this notion for and irreducible, automorphic, cuspidal
representation associated to $\GL(n)$. Under certain general
hypotheses, it is shown in this paper that every distinguished,
irreducible, supercuspidal representation may be realized as a
local component of a distinguished, irreducible automorphic,
cuspidal representation. Applications to the theory of
distinguished supercuspidal representations are provided.
Categories:22E50, 22E35, 11F70 |
145. CMB 2002 (vol 45 pp. 115)
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The Number of Non-Zero Digits of $n!$ Let $b$ be an integer with $b>1$. In this note, we prove that the
number of non-zero digits in the base $b$ representation of $n!$
grows at least as fast as a constant, depending on $b$, times $\log
n$.
Category:11A63 |
146. CMB 2002 (vol 45 pp. 109)
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The Distribution of Totatives D.~H.~Lehmer initiated the study of the distribution of totatives, which
are numbers coprime with a given integer. This led to various problems
considered by P.~Erd\H os, who made a conjecture on such distributions.
We prove his conjecture by establishing a theorem on the ordering of
residues.
Keywords:Euler's function, totatives Categories:11A05, 11A07, 11A25 |
147. CMB 2002 (vol 45 pp. 123)
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Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |
148. CMB 2002 (vol 45 pp. 97)
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Invariant Measures and Natural Extensions We study ergodic properties of a family of interval maps that are
given as the fractional parts of certain real M\"obius
transformations. Included are the maps that are exactly
$n$-to-$1$, the classical Gauss map and the Renyi or backward
continued fraction map. A new approach is presented for deriving
explicit realizations of natural automorphic extensions and their
invariant measures.
Keywords:Continued fractions, interval maps, invariant measures Categories:11J70, 58F11, 58F03 |
149. CMB 2002 (vol 45 pp. 86)
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On Cyclic Fields of Odd Prime Degree $p$ with Infinite Hilbert $p$-Class Field Towers Let $k$ be a cyclic extension of odd prime degree $p$ of the field of
rational numbers. If $t$ denotes the number of primes that ramify in $k$,
it is known that the Hilbert $p$-class field tower of $k$ is infinite if
$t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive
proportion of such fields $k$ have infinite Hilbert $p$-class field towers.
Categories:11R29, 11R37, 11R45 |
150. CMB 2002 (vol 45 pp. 36)
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Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$ Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains
$\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there
is a unique normalised generator of the field of $G$-automorphic
functions which is known as a normalised Hauptmodul. This paper gives
a characterisation of normalised Hauptmoduls as formal $q$ series
using modular polynomials.
Categories:11F03, 11F22, 30F35 |
