126. CMB 2006 (vol 49 pp. 472)
127. CMB 2006 (vol 49 pp. 428)
 Lee, Min Ho

VectorValued Modular Forms of Weight Two Associated With JacobiLike Forms
We construct vectorvalued modular forms of weight 2 associated to
Jacobilike forms with respect to a symmetric tensor representation of
$\G$ by using the method of Kuga and Shimura as well as the
correspondence between Jacobilike forms and sequences of modular forms.
As an application, we obtain vectorvalued modular forms determined by
theta functions and by pseudodifferential operators.
Categories:11F11, 11F50 

128. CMB 2006 (vol 49 pp. 196)
129. CMB 2006 (vol 49 pp. 296)
 Sch"utt, Matthias

On the Modularity of Three CalabiYau Threefolds With Bad Reduction at 11
This paper investigates the modularity of three
nonrigid CalabiYau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$adic cohomology groups are shown to split into
twodimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
twodimensional 2adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hessepencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 

130. CMB 2006 (vol 49 pp. 247)
131. CMB 2006 (vol 49 pp. 21)
 Chapman, Robin; Hart, William

Evaluation of the Dedekind Eta Function
We extend the methods of Van der Poorten and Chapman
for
explicitly evaluating the Dede\kind eta function at quadratic
irrationalities. Via evaluation of Hecke
$L$series we obtain new evaluations at points in
imaginary quadratic number fields with
class numbers 3 and 4. Further, we overcome the limitations
of the earlier methods and via modular equations provide
explicit evaluations where the class number is 5 or 7.
Category:11G15 

132. CMB 2006 (vol 49 pp. 108)
 Kwapisz, Jaroslaw

A Dynamical Proof of Pisot's Theorem
We give a geometric proof of classical results that characterize
Pisot numbers as algebraic $\lambda>1$ for which
there is $x\neq0$ with $\lambda^nx \to 0 \mod$ and identify such
$x$ as members of $\Z[\lambda^{1}] \cdot \Z[\lambda]^*$ where $\Z[\lambda]^*$ is the dual module of $\Z[\lambda]$.
Category:11R06 

133. CMB 2005 (vol 48 pp. 576)
 Ichimura, Humio

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II
Let $m=p^e$ be a power of a prime number $p$.
We say that a number field $F$ satisfies the property $(H_m')$
when for any $a \in F^{\times}$, the cyclic extension
$F(\z_m, a^{1/m})/F(\z_m)$ has a normal $p$integral basis.
We prove that $F$ satisfies $(H_m')$
if and only if the natural homomorphism $Cl_F' \to Cl_K'$ is trivial.
Here $K=F(\zeta_m)$, and $Cl_F'$ denotes the ideal class group of $F$
with respect to the $p$integer ring of $F$.
Category:11R33 

134. CMB 2005 (vol 48 pp. 535)
135. CMB 2005 (vol 48 pp. 636)
136. CMB 2005 (vol 48 pp. 428)
137. CMB 2005 (vol 48 pp. 333)
 Alzer, Horst

Monotonicity Properties of the Hurwitz Zeta Function
Let
$$
\zeta(s,x)=\sum_{n=0}^{\infty}\frac{1}{(n+x)^s} \quad{(s>1,\, x>0)}
$$
be the Hurwitz zeta function and let
$$
Q(x)=Q(x;\alpha,\beta;a,b)=\frac{(\zeta(\alpha,x))^a}{(\zeta(\beta,x))^b},
$$
where $\alpha, \beta>1$
and $a,b>0$ are real numbers. We prove:
(i) The function $Q$ is decreasing on $(0,\infty)$ if{}f $\alpha a\beta b\geq \max(ab,0)$.
(ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a\beta b\leq
\min(ab,0)$.
An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s1)\zeta(s,x)]^{1/(s1)}$ is decreasing on $(1,\infty)$. This settles
a conjecture of Bastien and Rogalski.
Categories:11M35, 26D15 

138. CMB 2005 (vol 48 pp. 394)
 Đoković, D. Ž.; Szechtman, F.; Zhao, K.

Diagonal Plus Tridiagonal Representatives for Symplectic Congruence Classes of Symmetric Matrices
Let $n=2m$ be even and denote by $\Sp_n(F)$ the symplectic group
of rank $m$ over an infinite field $F$ of characteristic different
from $2$. We show that any $n\times n$ symmetric matrix $A$ is
equivalent under symplectic congruence transformations to the
direct sum of $m\times m$ matrices $B$ and $C$, with $B$ diagonal
and $C$ tridiagonal. Since the $\Sp_n(F)$module of symmetric
$n\times n$ matrices over $F$ is isomorphic to the adjoint module
$\sp_n(F)$, we infer that any adjoint orbit of $\Sp_n(F)$ in
$\sp_n(F)$ has a representative in the sum of $3m1$ root spaces,
which we explicitly determine.
Categories:11E39, 15A63, 17B20 

139. CMB 2005 (vol 48 pp. 211)
 Germain, Jam

The Distribution of Totatives
The integers coprime to $n$ are called the {\it totatives} \rm of $n$.
D. H. Lehmer and Paul Erd\H{o}s were interested in understanding when
the number of totatives between $in/k$ and $(i+1)n/k$ are $1/k$th of
the total number of totatives up to $n$. They provided criteria in
various cases. Here we give an ``if and only if'' criterion which
allows us to recover most of the previous results in this literature
and to go beyond, as well to reformulate the problem in terms of
combinatorial group theory. Our criterion is that the above holds if
and only if for every odd character $\chi \pmod \kappa$ (where
$\kappa:=k/\gcd(k,n/\prod_{pn} p)$) there exists a prime $p=p_\chi$
dividing $n$ for which $\chi(p)=1$.
Categories:11A05, 11A07, 11A25, 20C99 

140. 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 

141. CMB 2005 (vol 48 pp. 16)
 Cojocaru, Alina Carmen

On the Surjectivity of the Galois Representations Associated to NonCM Elliptic Curves
Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {\emph{not}} surjective.
Categories:11G05, 11N36, 11R45 

142. CMB 2005 (vol 48 pp. 121)
143. CMB 2004 (vol 47 pp. 589)
144. CMB 2004 (vol 47 pp. 573)
145. 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 

146. 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 

147. 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 

148. 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 

149. CMB 2004 (vol 47 pp. 468)
150. 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 
