151. 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 

152. CMB 2003 (vol 46 pp. 71)
153. 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 

154. 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 

155. 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 

156. 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 

157. 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 

158. 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 

159. CMB 2002 (vol 45 pp. 220)
 Hakim, Jeffrey; Murnaghan, Fiona

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 

160. CMB 2002 (vol 45 pp. 257)
 Lee, Min Ho

Modular Forms Associated to Theta Functions
We use the theory of Jacobilike 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 

161. 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 

162. 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 

163. 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 

164. 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 

165. CMB 2002 (vol 45 pp. 138)
166. CMB 2002 (vol 45 pp. 36)
167. CMB 2002 (vol 45 pp. 109)
 Hall, R. R.; Shiu, P.

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 

168. CMB 2002 (vol 45 pp. 115)
 Luca, Florian

The Number of NonZero Digits of $n!$
Let $b$ be an integer with $b>1$. In this note, we prove that the
number of nonzero digits in the base $b$ representation of $n!$
grows at least as fast as a constant, depending on $b$, times $\log
n$.
Category:11A63 

169. CMB 2002 (vol 45 pp. 86)
 Gerth, Frank

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 

170. CMB 2002 (vol 45 pp. 97)
 Haas, Andrew

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 

171. CMB 2002 (vol 45 pp. 123)
 Moody, Robert V.

Uniform Distribution in Model Sets
We give a new measuretheoretical 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 

172. CMB 2001 (vol 44 pp. 385)
 Ballantine, Cristina M.

A Hypergraph with Commuting Partial Laplacians
Let $F$ be a totally real number field and let $\GL_{n}$ be the
general linear group of rank $n$ over $F$. Let $\mathfrak{p}$
be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$
with respect to the valuation induced by $\mathfrak{p}$. We will
consider a finite quotient of the affine building of the group
$\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object
as a hypergraph and find a set of commuting operators whose sum will
be the usual adjacency operator of the graph underlying the hypergraph.
Keywords:Hecke operators, buildings Categories:11F25, 20F32 

173. CMB 2001 (vol 44 pp. 398)
 Cardon, David A.; Ram Murty, M.

Exponents of Class Groups of Quadratic Function Fields over Finite Fields
We find a lower bound on the number of imaginary quadratic extensions
of the function field $\F_q(T)$ whose class groups have an element of
a fixed order.
More precisely, let $q \geq 5$ be a power of an odd prime and let $g$
be a fixed positive integer $\geq 3$. There are $\gg q^{\ell
(\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D)
\leq \ell$ such that the class groups of the quadratic extensions
$\F_q(T,\sqrt{D})$ have an element of order~$g$.
Keywords:class number, quadratic function field Categories:11R58, 11R29 

174. CMB 2001 (vol 44 pp. 440)
 Hironaka, Eriko

The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the wellknown Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 

175. CMB 2001 (vol 44 pp. 282)
 Lee, Min Ho; Myung, Hyo Chul

Hecke Operators on Jacobilike Forms
Jacobilike forms for a discrete subgroup $\G \subset \SL(2,\mbb R)$
are formal power series with coefficients in the space of functions on
the Poincar\'e upper half plane satisfying a certain functional
equation, and they correspond to sequences of certain modular forms.
We introduce Hecke operators acting on the space of Jacobilike forms
and obtain an explicit formula for such an action in terms of modular
forms. We also prove that those Hecke operator actions on Jacobilike
forms are compatible with the usual Hecke operator actions on modular
forms.
Categories:11F25, 11F12 
