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126. 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.


127. CMB 2004 (vol 47 pp. 431)

Osburn, Robert
A Note on $4$-Rank Densities
For certain real quadratic number fields, we prove density results concerning $4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol.

Categories:11R70, 19F99, 11R11, 11R45

128. CMB 2004 (vol 47 pp. 468)

Soundararajan, K.
Strong Multiplicity One for the Selberg Class
We investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.

Categories:11M41, 11M26, 11M06

129. CMB 2004 (vol 47 pp. 237)

Laubie, François
Ramification des séries 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: $$ u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} + \cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in \mathbb{N}). $$ 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

130. 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

131. CMB 2004 (vol 47 pp. 271)

Naumann, Niko
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms
We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$.

Categories:11G10, 14K15

132. 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 so-called ``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

133. CMB 2003 (vol 46 pp. 546)

Long, Ling
$L$-Series of Certain Elliptic Surfaces
In this paper, we study the modularity of certain elliptic surfaces by determining their $L$-series through their monodromy groups.

Categories:14J27, 11M06

134. 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

135. 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.


136. 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

137. CMB 2003 (vol 46 pp. 178)

Jaulent, Jean-François; 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

138. CMB 2003 (vol 46 pp. 229)

Lin, Ke-Pao; 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 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

139. 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

140. 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.


141. CMB 2003 (vol 46 pp. 157)

Wieczorek, Małgorzata
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.


142. 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

143. CMB 2003 (vol 46 pp. 26)

Bernardi, D.; Halberstadt, E.; Kraus, A.
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 $\mathbb{P}^3$.


144. 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

145. 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

146. 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

147. 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

148. 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

149. 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

150. 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
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