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101. CMB 2008 (vol 51 pp. 172)

Alkan, Emre; Zaharescu, Alexandru
 Consecutive Large Gaps in Sequences Defined by Multiplicative Constraints In this paper we obtain quantitative results on the occurrence of consecutive large gaps between $B$-free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication. Keywords:$B$-free numbers, consecutive gapsCategories:11N25, 11B05

102. CMB 2008 (vol 51 pp. 3)

 Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.

103. CMB 2008 (vol 51 pp. 57)

Dobrowolski, Edward
 A Note on Integer Symmetric Matrices and Mahler's Measure We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick. Keywords:integer matrices, Lehmer's problem, Mahler's measureCategories:11C20, 11R06

104. CMB 2008 (vol 51 pp. 32)

Choi, Stephen; Zhou, Ping
 On Linear Independence of a Certain Multivariate Infinite Product Let $q,m,M \ge 2$ be positive integers and $r_1,r_2,\dots ,r_m$ be positive rationals and consider the following $M$ multivariate infinite products $F_i = \prod_{j=0}^\infty ( 1+q^{-(Mj+i)}r_1+q^{-2(Mj+i)}r_2+\dots + q^{-m(Mj+i)}r_m)$ for $i=0,1,\dots ,M-1$. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space $\IQ F_0+\IQ F_1 +\dots + \IQ F_{M-1} + \IQ$ over $\IQ$ and show that among these $M$ infinite products, $F_0, F_1,\dots ,F_{M-1}$, at least $\sim M/m(m+1)$ of them are irrational for fixed $m$ and $M \rightarrow \infty$. Category:11J72

105. CMB 2008 (vol 51 pp. 134)

Rosales, J. C.; Garc\'{\i}a-Sánchez, P. A.
 Numerical Semigroups Having a Toms Decomposition We show that the class of system proportionally modular numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. Categories:20M14, 11D75

106. CMB 2008 (vol 51 pp. 100)

Petkov, Vesselin
 Dynamical Zeta Function for Several Strictly Convex Obstacles The behavior of the dynamical zeta function $Z_D(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function $\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re s = s_2$, where $s_2$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$. Keywords:dynamical zeta function, periodic raysCategories:11M36, 58J50

107. CMB 2007 (vol 50 pp. 486)

Cynk, S.; Hulek, K.
 Higher-Dimensional Modular\\Calabi--Yau Manifolds We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity. Categories:14G10, 14J32, 11G40

108. CMB 2007 (vol 50 pp. 594)

Laubie, François
 Ramification des groupes abÃ©liens d'automorphismes des corps $\mathbb F_q(\!(X)\!)$ Soit $q$ une puissance d'un nombre premier $p$. Dans cette note on \'etablit la g\'en\'eralisation suivante d'un th\'eor\eme de Wintenberger : tout sous-groupe ab\'elien ferm\'e du groupe des $\mathbb F_q$-auto\-morphismes continus du corps des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration de ramification est un groupe filtr\'e isomorphe au groupe de Galois d'une extension ab\'elienne d'un corps local {\a} corps r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification de l'extension en num\'erotation inf\'erieure. Category:11S15

109. CMB 2007 (vol 50 pp. 399)

Komornik, Vilmos; Loreti, Paola
 Expansions in Complex Bases Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Dar\'oczy and K\'atai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems. Keywords:non-integer bases, greedy expansions, beta-expansionsCategories:11A67, 11A63, 11B85

110. CMB 2007 (vol 50 pp. 334)

Chiang-Hsieh, Hung-Jen; Yang, Yifan
 Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is a prime and $w_p$ is the Atkin--Lehner involution. We then use the Hauptmoduls, along with modular functions on $\Gamma_1(p)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given. Categories:11F03, 11G16, 11R20

111. CMB 2007 (vol 50 pp. 409)

Luca, Florian; Shparlinski, Igor E.
 Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$. Categories:11G20, 11N32, 11R11

112. 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 integralCategories:11B65, 33D15

113. CMB 2007 (vol 50 pp. 215)

Kloosterman, Remke
 Elliptic $K3$ Surfaces with Geometric Mordell--Weil Rank $15$ We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$. Categories:14J27, 14J28, 11G05

114. CMB 2007 (vol 50 pp. 234)

Kuo, Wentang
 A Remark on a Modular Analogue of the Sato--Tate Conjecture The original Sato--Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate a modular analogue of the Sato--Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate--Tate measure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform. Keywords:$L$-functions, Elliptic curves, Sato--TateCategories:11F03, 11F25

115. CMB 2007 (vol 50 pp. 313)

Tzermias, Pavlos
 On Cauchy--Liouville--Mirimanoff Polynomials Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy--Liouville--Mirimanoff polynomials to show that the intersection of the Fermat curve of degree $p$ with the line $X+Y=Z$ in the projective plane contains no algebraic points of degree $d$ with $3 \leq d \leq 11$. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of P\'{o}lya and Szeg\"{o}. These conditions are \emph{conjecturally} also necessary for irreducibility. Categories:11G30, 11R09, 12D05, 12E10

116. CMB 2007 (vol 50 pp. 191)

Drungilas, Paulius; Dubickas, Artūras
 Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$, say $d$, one of these two polynomials is irreducible and another has an irreducible factor of degree $d$, so that $\alpha=M(P)-bM(Q)$ with irreducible polynomials $P, Q\in \mathbb Z[X]$ of degree $d$ and a positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$. Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjectureCategories:11R04, 11R06, 11R09, 11R33, 11D09

117. CMB 2007 (vol 50 pp. 196)

Fernández, Julio; González, Josep; Lario, Joan-C.
 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 genus-three case $N=5$. Categories:11F03, 11F80, 14G05

118. 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 $$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1} x^{\phi(m)-1} + \dots + c_{\phi(m)},$$ 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

119. 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 $\sum_{j=1}^{n-1}j^{n-1}\equiv -1 \tmod{n}$. Then $G(X)\ll X^{1/2}\log X$ for sufficiently large $X$. Category:11A51

120. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
 van der Pol Expansions of L-Series We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series. Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-seriesCategories:11M35, 11M41, 30B50

121. 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 curveCategories:11B83, 11Y85, 11D09

122. 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

123. 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

124. 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 NÃ©ron-Severi group of this surface and find all low degree curves on it. Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theoryCategories:14G05, 14J28, 11D41

125. 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
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