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51. CMB 2011 (vol 56 pp. 283)

Coons, Michael
 Transcendental Solutions of a Class of Minimal Functional Equations We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ satisfying a functional equation of the form $$f(z^d)=\sum_{k=1}^n \frac{A_k(z)}{B_k(z)}f(z)^k,$$ where $A_k(z),B_k(z)\in \mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a minimal functional equation of the above form with $n\geqslant 2$, then $f(z)$ is necessarily transcendental. Towards a more complete classification, the case $n=1$ is also considered. Keywords:transcendence, generating functions, Mahler-type functional equationCategories:11B37, 11B83, , 11J91

52. CMB 2011 (vol 56 pp. 70)

Hrubeš, P.; Wigderson, A.; Yehudayoff, A.
 An Asymptotic Bound on the Composition Number of Integer Sums of Squares Formulas Let $\sigma_{\mathbb Z}(k)$ be the smallest $n$ such that there exists an identity $(x_1^2 + x_2^2 + \cdots + x_k^2) \cdot (y_1^2 + y_2^2 + \cdots + y_k^2) = f_1^2 + f_2^2 + \cdots + f_n^2,$ with $f_1,\dots,f_n$ being polynomials with integer coefficients in the variables $x_1,\dots,x_k$ and $y_1,\dots,y_k$. We prove that $\sigma_{\mathbb Z}(k) \geq \Omega(k^{6/5})$. Keywords:composition formulas, sums of squares, Radon-Hurwitz numberCategory:11E25

53. CMB 2011 (vol 55 pp. 850)

Shparlinski, Igor E.; Stange, Katherine E.
 Character Sums with Division Polynomials We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author. Keywords:division polynomial, character sumCategories:11L40, 14H52

54. CMB 2011 (vol 55 pp. 774)

Mollin, R. A.; Srinivasan, A.
 Pell Equations: Non-Principal Lagrange Criteria and Central Norms We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D>1$. We also provide a simple criterion for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of congruence conditions modulo $D$. Keywords:Pell's equation, continued fractions, central normsCategories:11D09, 11A55, 11R11, 11R29

55. CMB 2011 (vol 55 pp. 435)

Zelator, Konstantine
 A Note on the Diophantine Equation $x^2 + y^6 = z^e$, $e \geq 4$ We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$. We show that, when $e$ is a multiple of $4$ or $6$, this equation has no solutions in positive integers with $x$ and $y$ relatively prime. As a corollary, we show that there exists no primitive Pythagorean triangle one of whose leglengths is a perfect cube, while the hypotenuse length is an integer square. Keywords:diophantine equationCategory:11D

56. CMB 2011 (vol 55 pp. 400)

Sebbar, Abdellah; Sebbar, Ahmed
 Eisenstein Series and Modular Differential Equations The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms. Keywords:differential equations, modular forms, Schwarz derivative, equivariant formsCategories:11F11, 34M05

57. CMB 2011 (vol 55 pp. 26)

Bertin, Marie José
 A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series. Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen bracketsCategories:11, 14D, 14J

58. CMB 2011 (vol 55 pp. 67)

Cummins, C. J.; Duncan, J. F.
 An $E_8$ Correspondence for Multiplicative Eta-Products We describe an $E_8$ correspondence for the multiplicative eta-products of weight at least $4$. Keywords:We describe an E8 correspondence for the multiplicative eta-products of weight at leastÂ 4.Categories:11F20, 11F12, 17B60

59. CMB 2011 (vol 55 pp. 193)

Ulas, Maciej
 Rational Points in Arithmetic Progressions on $y^2=x^n+k$ Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$ for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$ with the property that on the elliptic curve $\mathcal{E}': y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\mathcal{E}'$. In particular this result generalizes earlier results of Lee and V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves $y^2=x^n+k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves $y^2=x^n+k$ there are six rational points in arithmetic progression. Keywords:arithmetic progressions, elliptic curves, rational points on hyperelliptic curvesCategory:11G05

60. CMB 2011 (vol 54 pp. 748)

Shparlinski, Igor E.
 On the Distribution of Irreducible Trinomials We obtain new results about the number of trinomials $t^n + at + b$ with integer coefficients in a box $(a,b) \in [C, C+A] \times [D, D+B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most $p^{1/2+o(1)}$, improving on the previous estimate of $p^{2/3+o(1)}$ obtained by the author in 1989. Keywords:irreducible trinomials, character sumsCategories:11L40, 11T06

61. CMB 2011 (vol 55 pp. 38)

Butske, William
 Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras Zarhin proves that if $C$ is the curve $y^2=f(x)$ where $\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his result in the genus $g=2$ case supposing other Galois groups, we calculate $\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$ for a genus $2$ curve where $f(x)$ is irreducible. In particular, we show that unless the Galois group is $S_5$ or $A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$. Categories:11G10, 20C20

62. CMB 2011 (vol 54 pp. 645)

Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega
 An Extension of Craig's Family of Lattices Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties. Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fieldsCategories:11H31, 11H55, 11H50, 11R18, 11R04

63. CMB 2011 (vol 55 pp. 60)

Coons, Michael
 Extension of Some Theorems of W. Schwarz In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C} [\mkern-3mu[ z]\mkern-3mu]$ satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$ with $A(z)\neq 0$ and $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 64. CMB 2011 (vol 54 pp. 757) Sun, Qingfeng  Cancellation of Cusp Forms Coefficients over Beatty Sequences on$\textrm{GL}(m)$Let$A(n_1,n_2,\dots,n_{m-1})$be the normalized Fourier coefficients of a Maass cusp form on$\textrm{GL}(m)$. In this paper, we study the cancellation of$A (n_1,n_2,\dots,n_{m-1})$over Beatty sequences. Keywords:Fourier coefficients, Maass cusp form on$\textrm{GL}(m)$, Beatty sequenceCategories:11F30, 11M41, 11B83 65. CMB 2011 (vol 54 pp. 739) Samuels, Charles L.  The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number$\alpha$by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure. Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problemCategories:11R04, 11R09 66. CMB 2011 (vol 54 pp. 316) Mazhouda, Kamel  The Saddle-Point Method and the Li Coefficients In this paper, we apply the saddle-point method in conjunction with the theory of the NÃ¶rlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function$F$in the Selberg class$\mathcal{S}$and under the Generalized Riemann Hypothesis, we have $$\lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),$$ with $$c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}},$$ where$\gamma$is the Euler's constant and the notation is as below. Keywords:Selberg class, Saddle-point method, Riemann Hypothesis, Li's criterionCategories:11M41, 11M06 67. CMB 2011 (vol 54 pp. 288) Jacobs, David P.; Rayes, Mohamed O.; Trevisan, Vilmar  The Resultant of Chebyshev Polynomials Let$T_{n}$denote the$n$-th Chebyshev polynomial of the first kind, and let$U_{n}$denote the$n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant$\operatorname{res}( T_{m}, T_{n} )$. Similarly, we give a formula for$\operatorname{res}( U_{m}, U_{n} )$. Keywords:resultant, Chebyshev polynomialCategories:11Y11, 68W20 68. CMB 2011 (vol 54 pp. 330) Mouhib, A.  Sur la borne infÃ©rieure du rang du 2-groupe de classes de certains corps multiquadratiques Soient$p_1,p_2,p_3$et$q$des nombres premiers distincts tels que$p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$,$k = \mathbf{Q} (\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$et$\operatorname{Cl}_2(k)$le$2$-groupe de classes de$k$. A. FrÃ¶hlich a dÃ©montrÃ© que$\operatorname{Cl}_2(k)$n'est jamais trivial. Dans cet article, nous donnons une extension de ce rÃ©sultat, en dÃ©montrant que le rang de$\operatorname{Cl}_2(k)$est toujours supÃ©rieur ou Ã©gal Ã$2$. Nous dÃ©montrons aussi, que la valeur$2$est optimale pour une famille infinie de corps$k$. Keywords:class group, units, multiquadratic number fieldsCategories:11R29, 11R11 69. CMB 2010 (vol 54 pp. 39) Chapman, S. T.; García-Sánchez, P. A.; Llena, D.; Marshall, J.  Elements in a Numerical Semigroup with Factorizations of the Same Length Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions. Keywords:numerical monoid, numerical semigroup, non-unique factorizationCategories:20M14, 20D60, 11B75 70. CMB 2010 (vol 53 pp. 661) Johnstone, Jennifer A.; Spearman, Blair K.  Congruent Number Elliptic Curves with Rank at Least Three We give an infinite family of congruent number elliptic curves each with rank at least three. Keywords:congruent number, elliptic curve, rankCategory:11G05 71. CMB 2010 (vol 53 pp. 654) Elliott, P. D. T. A.  Variations on a Paper of ErdÅs and Heilbronn It is shown that an old direct argument of ErdÅs and Heilbronn may be elaborated to yield a result of the current inverse type. Categories:11L07, 11P70 72. CMB 2010 (vol 53 pp. 385) Achter, Jeffrey D.  Exceptional Covers of Surfaces Consider a finite morphism$f: X \rightarrow Y$of smooth, projective varieties over a finite field$\mathbf{F}$. Suppose$X$is the vanishing locus in$\mathbf{P}^N$of$r$forms of degree at most$d$. We show that there is a constant$C$depending only on$(N,r,d)$and$\deg(f)$such that if$|{\mathbf{F}}|>C$, then$f(\mathbf{F}): X(\mathbf{F}) \rightarrow Y(\mathbf{F})$is injective if and only if it is surjective. Category:11G25 73. CMB 2010 (vol 53 pp. 571) Trifković, Mak  Periods of Modular Forms and Imaginary Quadratic Base Change Let$f$be a classical newform of weight$2$on the upper half-plane$\mathcal H^{(2)}$,$E$the corresponding strong Weil curve,$K$a class number one imaginary quadratic field, and$F$the base change of$f$to$K$. Under a mild hypothesis on the pair$(f,K)$, we prove that the period ratio$\Omega_E/(\sqrt{|D|}\Omega_F)$is in$\mathbb Q$. Here$\Omega_F$is the unique minimal positive period of$F$, and$\Omega_E$the area of$E(\mathbb C)\$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley. Category:11F67

74. CMB 2009 (vol 53 pp. 87)

Ghioca, Dragos
 Elliptic Curves over the Perfect Closure of a Function Field We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated. Keywords:elliptic curves, heightsCategories:11G50, 11G05

75. CMB 2009 (vol 53 pp. 204)

Alkan, Emre; Zaharescu, Alexandru
 Corrigendum for "Consecutive large gaps in sequences defined by multiplicative constraints" No abstract. Categories:11N25, 11B05
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