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126. CMB 2005 (vol 48 pp. 147)

 Baker-Type Estimates for Linear Forms in the Values of $q$-Series We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field~$\II$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincar\'e-type equations and the connection between the solutions of these functional equations and the generalized Heine series. Keywords:measure of linear independence, $q$-seriesCategories:11J82, 33D15

127. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
 Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc$ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$. Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fieldsCategories:11A55, 11D09, 11R11

128. CMB 2004 (vol 47 pp. 589)

Liu, Yu-Ru
 A Generalization of the ErdÃ¶s-Kac Theorem and its Applications We axiomatize the main properties of the classical Erd\"os-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field. Categories:11N60, 11N80

129. CMB 2004 (vol 47 pp. 573)

Liu, Yu-Ru
 A Generalization of the TurÃ¡n Theorem and Its Applications We axiomatize the main properties of the classical TurÃ¡n Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field. Categories:11N37, 11N80

130. CMB 2004 (vol 47 pp. 373)

Győry, K.; Hajdu, L.; Saradha, N.
 On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obl\'ath for the case of squares, and an extension of a theorem of Gy\H ory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in $n,y$ when $b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$ when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$. Category:11D41

131. 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

132. CMB 2004 (vol 47 pp. 398)

McKinnon, David
 A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating rational curves such that the density of rational points on $U-Z_U$ is strictly less than the density of rational points on $Z_U$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets $Z_U$ for successively smaller sets $U$. In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, heightCategories:11G35, 14G05

133. 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

134. 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

135. 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: $$u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} + \cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in \mathbb{N}).$$ 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 groupCategories:11S15, 20E18

136. 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

137. 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

138. 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

139. 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 $$N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q) Categories:11G50, 14J28, 14G40, 14J50, 14G05 140. CMB 2003 (vol 46 pp. 546) Long, Ling  L-Series of Certain Elliptic Surfaces In this paper, we study the modularity of certain elliptic surfaces by determining their L-series through their monodromy groups. Categories:14J27, 11M06 141. CMB 2003 (vol 46 pp. 344) Gurak, S.  Gauss and Eisenstein Sums of Order Twelve Let q=p^{r} with p an odd prime, and \mathbf{F}_{q} denote the finite field of q elements. Let \Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p}  be the usual trace map and set \zeta_{p} =\exp(2\pi i/p). For any positive integer e, define the (modified) Gauss sum g_{r}(e) by$$ 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

142. 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

143. 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

144. 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

145. 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

146. 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

147. 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

148. 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

149. 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. Category:11G05

150. 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
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