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101. 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

102. CMB 2006 (vol 49 pp. 578)

Muić, Goran
On the Structure of the Full Lift for the Howe Correspondence of $(Sp(n), O(V))$ for Rank-One Reducibilities
In this paper we determine the structure of the full lift for the Howe correspondence of $(Sp(n),O(V))$ for rank-one reducibilities.

Categories:22E35, 22E50, 11F70

103. 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 non-Kummer, singular K3 surface are dense. We will also compute the entire Néron-Severi group of this surface and find all low degree curves on it.

Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, Néron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory
Categories:14G05, 14J28, 11D41

104. 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

105. CMB 2006 (vol 49 pp. 428)

Lee, Min Ho
Vector-Valued Modular Forms of Weight Two Associated With Jacobi-Like Forms
We construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of $\G$ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.

Categories:11F11, 11F50

106. CMB 2006 (vol 49 pp. 472)

Silvester, Alan K.; Spearman, Blair K.; Williams, Kenneth S.
Cyclic Cubic Fields of Given Conductor and Given Index
The number of cyclic cubic fields with a given conductor and a given index is determined.

Keywords:Discriminant, conductor, index, cyclic cubic field
Categories:11R16, 11R29

107. CMB 2006 (vol 49 pp. 448)

Pacelli, Allison M.
A Lower Bound on the Number of Cyclic Function Fields With Class Number Divisible by $n$
In this paper, we find a lower bound on the number of cyclic function fields of prime degree~$l$ whose class numbers are divisible by a given integer $n$. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible by $n$.

Categories:11R29, 11R58

108. CMB 2006 (vol 49 pp. 196)

Chernousov, Vladimir
Another Proof of Totaro's Theorem on $E_8$-Torsors
We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$.

Categories:11E72, 14M17, 20G15

109. CMB 2006 (vol 49 pp. 296)

Sch"utt, Matthias
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11
This paper investigates the modularity of three non-rigid Calabi--Yau 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 two-dimensional 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 two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

Categories:14J32, 11F11, 11F23, 20C12

110. CMB 2006 (vol 49 pp. 247)

Myjak, Józef; Szarek, Tomasz; Ślȩczka, Maciej
A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces
Let $X$ be a Polish space. We will prove that $$ \dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to } X\}, $$ where $\dim_L X$ and $\dim_T X$ stand for the concentration dimension and the topological dimension of $X$, respectively.

Keywords:Hausdorff dimension, topological dimension, Lévy concentration function, concentration dimension
Categories:11K55, 28A78

111. 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

112. 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

113. 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

114. CMB 2005 (vol 48 pp. 535)

Ellenberg, Jordan S.
On the Error Term in Duke's Estimate for the Average Special Value of $L$-Functions
Let $\FF$ be an orthonormal basis for weight $2$ cusp forms of level $N$. We show that various weighted averages of special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4 \pi c + O(N^{-1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error term of $O(N^{-1/2}\log N)$.

Categories:11F67, 11F11

115. CMB 2005 (vol 48 pp. 636)

Győry, K.; Hajdu, L.; Saradha, N.
Correction to: On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$
In the article under consideration (Canad. Math. Bull. \textbf{47} (2004), pp.~373--388), Lemma 6 is not true in the form presented there. Lemma 6 is used only in the proof of part (i) of Theorem 9. We note, however, that part (i) of Theorem 9 in question is a special case of a theorem by Bennet, Bruin, Gy\H{o}ry and Hajdu.

Category:11D41

116. CMB 2005 (vol 48 pp. 428)

Miyamoto, Roland; Top, Jaap
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2)
and fibre type for elliptic curves over discrete valued fields of equal characteristic~3. Along the same lines, partial results are obtained in equal characteristic~2.

Categories:14H52, 14K15, 11G07, 11G05, 12J10

117. 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 $\alpha a-\beta b\geq \max(a-b,0)$. (ii) $Q$ is increasing on $(0,\infty)$ if{}f $\alpha a-\beta b\leq \min(a-b,0)$. An application of part (i) reveals that for all $x>0$ the function $s\mapsto [(s-1)\zeta(s,x)]^{1/(s-1)}$ is decreasing on $(1,\infty)$. This settles a conjecture of Bastien and Rogalski.

Categories:11M35, 26D15

118. 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 $3m-1$ root spaces, which we explicitly determine.

Categories:11E39, 15A63, 17B20

119. 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 $\kappa:=k/\gcd(k,n/\prod_{p|n} p)$) there exists a prime $p=p_\chi$ dividing $n$ for which $\chi(p)=1$.

Categories:11A05, 11A07, 11A25, 20C99

120. CMB 2005 (vol 48 pp. 147)

Väänänen, Keijo; Zudilin, Wadim
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, $q$-series
Categories:11J82, 33D15

121. 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 fields
Categories:11A55, 11D09, 11R11

122. CMB 2005 (vol 48 pp. 16)

Cojocaru, Alina Carmen
On the Surjectivity of the Galois Representations Associated to Non-CM 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

123. CMB 2004 (vol 47 pp. 573)

Liu, Yu-Ru
A Generalization of the Turán Theorem and Its Applications
We axiomatize the main properties of the classical Turán 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

124. CMB 2004 (vol 47 pp. 589)

Liu, Yu-Ru
A Generalization of the Erdös-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

125. 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
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