126. CMB 2005 (vol 48 pp. 147)
 Väänänen, Keijo; Zudilin, Wadim

BakerType 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 nonzero 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\'etype equations
and the connection between the solutions of these functional
equations and the generalized Heine series.
Keywords:measure of linear independence, $q$series Categories:11J82, 33D15 

127. CMB 2005 (vol 48 pp. 121)
128. CMB 2004 (vol 47 pp. 589)
129. CMB 2004 (vol 47 pp. 573)
130. CMB 2004 (vol 47 pp. 373)
 Győry, K.; Hajdu, L.; Saradha, N.

On the Diophantine Equation $n(n+d)\cdots(n+(k1)d)=by^l$
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 BatyrevManin Conjecture for Kummer Surfaces
Let $V$ be a $K3$ surface defined over a number field $k$. The
BatyrevManin 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 $UZ_U$ 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 BatyrevManin conjecture for $V$ can be reduced to the
BatyrevManin 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, BatyrevManin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 

133. CMB 2004 (vol 47 pp. 468)
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:
$$
u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma)  i_0(\gamma)}{p} +
\cdots + \frac{i_m (\gamma)  i_{m1} (\gamma)}{p^m} , \quad (m \in
\mathbb{N}).
$$
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 

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)
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 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 

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)
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, 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 

144. 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 

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 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 

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)
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)
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 
