101. CMB 2007 (vol 50 pp. 196)
 Fernández, Julio; González, Josep; Lario, JoanC.

Plane Quartic Twists of $X(5,3)$
Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genusthree case $N=5$.
Categories:11F03, 11F80, 14G05 

102. CMB 2007 (vol 50 pp. 284)
 McIntosh, Richard J.

Second Order Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $q<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
Keywords:$q$series, mock theta function, Mordell integral Categories:11B65, 33D15 

103. CMB 2007 (vol 50 pp. 71)
 Gurak, S.

Polynomials for Kloosterman Sums
Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$
denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman
sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq
m$, $(d,m)=1$,
satisfy the polynomial
$$f_{m}(x) = \prod_{d} (xR(d)) = x^{\phi(m)} +c_{1}
x^{\phi(m)1} + \dots + c_{\phi(m)},$$
where the sum and product are taken over a complete system of reduced residues
modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely,
$$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$
where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$
of reduced residues modulo $m$. Questions concerning the explicit
determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their
beginning coefficients), their reducibility over the rational field
${\bf Q}$ and duplication among the factors are studied. The treatment
is similar to what has been done for period polynomials for finite
fields.
Categories:11L05, 11T24 

104. CMB 2007 (vol 50 pp. 158)
 Tipu, Vicentiu

A Note on Giuga's Conjecture
Let $G(X)$ denote the number of positive composite integers $n$
satisfying $\sum_{j=1}^{n1}j^{n1}\equiv 1 \tmod{n}$.
Then $G(X)\ll X^{1/2}\log X$ for sufficiently large $X$.
Category:11A51 

105. CMB 2007 (vol 50 pp. 11)
 Borwein, David; Borwein, Jonathan

van der Pol Expansions of LSeries
We provide concise series representations for various
Lseries integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying Lseries.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, Lseries Categories:11M35, 11M41, 30B50 

106. CMB 2006 (vol 49 pp. 578)
107. CMB 2006 (vol 49 pp. 560)
 Luijk, Ronald van

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

108. CMB 2006 (vol 49 pp. 481)
 Browkin, J.; Brzeziński, J.

On Sequences of Squares with Constant Second Differences
The aim of this paper is to study sequences of integers
for which the second differences between their squares are
constant. We show that there are infinitely many nontrivial
monotone sextuples having this property and discuss some related
problems.
Keywords:sequence of squares, second difference, elliptic curve Categories:11B83, 11Y85, 11D09 

109. CMB 2006 (vol 49 pp. 526)
 Choi, So Young

The Values of Modular Functions and Modular Forms
Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero
and $\Gamma$ be a subgroup of $\Gamma_0$
of finite index of genus zero. We find universal recursive
relations giving the $q_{r}$series coefficients of
$j_0$ by using those of the $q_{h_{s}}$series of $j$, where $j$ is
the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul
for $\Gamma_0$ without zeros on the complex upper half plane
$\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for
$q$series coefficients of any modular form on
$\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.
Categories:10D12, 11F11 

110. CMB 2006 (vol 49 pp. 472)
111. 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 

112. CMB 2006 (vol 49 pp. 448)
113. 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 

114. CMB 2006 (vol 49 pp. 247)
115. CMB 2006 (vol 49 pp. 196)
116. 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 

117. 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 

118. 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 

119. CMB 2005 (vol 48 pp. 535)
120. CMB 2005 (vol 48 pp. 636)
121. 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 

122. 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 

123. CMB 2005 (vol 48 pp. 428)
124. 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 

125. 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 
