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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 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

152. CMB 2003 (vol 46 pp. 71)

Cutter, Pamela; Granville, Andrew; Tucker, Thomas J.
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

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.


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, non-split 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 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

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), 1007--1047].

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), 440--451.

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 non-trivial solution. Proofs are short and rest heavily on results whose proofs required Wiles' deep machinery.

Keywords:Diophantine equations

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)

Spearman, Blair K.; Williams, Kenneth S.
The Discriminant of a Dihedral Quintic Field Defined by a Trinomial $X^5 + aX + b$
Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be a root of $X^5 + aX + b$. An explicit formula is given for the discriminant of $Q(\theta)$.

Keywords:dihedral quintic field, trinomial, discriminant
Categories:11R21, 11R29

166. CMB 2002 (vol 45 pp. 36)

Cummins, C. J.
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

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 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 $\log n$.


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 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

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 well-known 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 Jacobi-like Forms
Jacobi-like 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 Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

Categories:11F25, 11F12
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