151. CMB 2004 (vol 47 pp. 358)
 Ford, Kevin

A Strong Form of a Problem of R. L. Graham
If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum
of $a_i/\gcd(a_i,a_j)$ over $a_i,a_j\in A$. We show that if
$G(A)$ is not too much larger than $M$, then $A$ must have a
special structure.
Category:11A05 

152. CMB 2004 (vol 47 pp. 264)
 McKinnon, David

Counting Rational Points on Ruled Varieties
In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.
Categories:11G50, 11D45, 11D04, 14G05 

153. CMB 2004 (vol 47 pp. 271)
154. CMB 2004 (vol 47 pp. 237)
 Laubie, François

Ramification des sÃ©ries formelles
Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$.
The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitution
law $\circ $ sometimes called the Nottingham group of $k$; it is denoted by
$\mathcal{R}_k$. The ramification of one series $\gamma\in\mathcal{R}_k$ is
caracterized by its lower ramification numbers: $i_m(\gamma)=\ord_X
\bigl(\gamma^{p^m} (X)/X  1\bigr)$, as well as its upper ramification numbers:
$$
u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma)  i_0(\gamma)}{p} +
\cdots + \frac{i_m (\gamma)  i_{m1} (\gamma)}{p^m} , \quad (m \in
\mathbb{N}).
$$
By Sen's theorem, the $u_m(\gamma)$ are integers. In this paper, we determine
the sequences of integers $(u_m)$ for which there exists $\gamma\in\mathcal{R}_k$
such that $u_m(\gamma)=u_m$ for all integer $m \geq 0$.
Keywords:ramification, Nottingham group Categories:11S15, 20E18 

155. CMB 2004 (vol 47 pp. 12)
 Burger, Edward B.

On Newton's Method and Rational Approximations to Quadratic Irrationals
In 1988 Rieger exhibited a differentiable function having a zero at
the golden ratio\break
$(1+\sqrt5)/2$ for which when Newton's method for approximating
roots is applied with an initial value $x_0=0$, all approximates
are socalled ``best rational approximates''in this case, of the
form $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$th Fibonacci
number. Recently this observation was extended by Komatsu to the
class of all quadratic irrationals whose continued fraction
expansions have period length $2$. Here we generalize these
observations by producing an analogous result for all quadratic
irrationals and thus provide an explanation for these phenomena.
Categories:11A55, 11B37 

156. CMB 2003 (vol 46 pp. 546)
157. CMB 2003 (vol 46 pp. 495)
 Baragar, Arthur

Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two
Let $V$ be an algebraic K3 surface defined over a number field $K$.
Suppose $V$ has Picard number two and an infinite group of
automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we
introduce the notion of a vector height $\mathbf{h} \colon V \to
\Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical
vector height $\widehat{\mathbf{h}}$ with the following properties:
\begin{gather*}
\widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\
h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1),
\end{gather*}
where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of
$\sigma$ (the pullback of $\sigma^{1}$), and $h_D$ is a Weil
height associated to the divisor $D$. The bounded function implied
by the $O(1)$ does not depend on $P$. This allows us to attack
some arithmetic problems. For example, we show that the number of
rational points with bounded logarithmic height in an
$\mathcal{A}$orbit satisfies
$$
N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q)
Categories:11G50, 14J28, 14G40, 14J50, 14G05 

158. CMB 2003 (vol 46 pp. 473)
 Yeats, Karen

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 

159. CMB 2003 (vol 46 pp. 344)
 Gurak, S.

Gauss and Eisenstein Sums of Order Twelve
Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite
field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be
the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive
integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by
$$
g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}}
$$
Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of
$g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity
present in a previous evaluation. Here I generalize Evans' result to give
a complete determination of the sum $g_{r}(12)$.
Categories:11L05, 11T24 

160. CMB 2003 (vol 46 pp. 229)
 Lin, KePao; Yau, Stephen S.T.

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 nonintegral
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
nonintegral 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 

161. CMB 2003 (vol 46 pp. 178)
 Jaulent, JeanFrançois; Maire, Christian

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 

162. CMB 2003 (vol 46 pp. 149)
 Scherk, John

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 

163. CMB 2003 (vol 46 pp. 71)
164. CMB 2003 (vol 46 pp. 39)
 Bülow, Tommy

Power Residue Criteria for Quadratic Units and the Negative Pell Equation
Let $d>1$ be a squarefree 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 2class
group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.
Categories:11R11, 11R27 

165. CMB 2003 (vol 46 pp. 26)
 Bernardi, D.; Halberstadt, E.; Kraus, A.

Remarques sur les points rationnels des variÃ©tÃ©s 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 

166. CMB 2003 (vol 46 pp. 157)
167. CMB 2002 (vol 45 pp. 606)
 Gannon, Terry

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 

168. CMB 2002 (vol 45 pp. 466)
 Arthur, James

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 

169. CMB 2002 (vol 45 pp. 428)
 Mollin, R. A.

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 

170. CMB 2002 (vol 45 pp. 337)
 Chen, Imin

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, nonsplit Cartan Categories:11G05, 11F80 

171. CMB 2002 (vol 45 pp. 364)
 Deitmar, Anton

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 

172. CMB 2002 (vol 45 pp. 247)
 Kihel, O.; Levesque, C.

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
nontrivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.
Keywords:Diophantine equations Category:11D41 

173. CMB 2002 (vol 45 pp. 196)
 Dubickas, Artūras

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 

174. CMB 2002 (vol 45 pp. 231)
 Hironaka, Eriko

Erratum:~~The Lehmer Polynomial and Pretzel Links
Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

175. CMB 2002 (vol 45 pp. 168)
 Byott, Nigel P.; Elder, G. Griffith

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), 10071047].
Categories:11S15, 20C11 
