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51. 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

52. 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

53. 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

54. 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

55. 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

56. 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 $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 57. CMB 2011 (vol 54 pp. 757) Sun, Qingfeng  Cancellation of Cusp Forms Coefficients over Beatty Sequences on$\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)$, Beatty sequenceCategories:11F30, 11M41, 11B83 58. CMB 2011 (vol 54 pp. 739) Samuels, Charles L.  The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number$\alpha$by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure. Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problemCategories:11R04, 11R09 59. CMB 2011 (vol 54 pp. 288) Jacobs, David P.; Rayes, Mohamed O.; Trevisan, Vilmar  The Resultant of Chebyshev Polynomials Let$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} )$. Keywords:resultant, Chebyshev polynomialCategories:11Y11, 68W20 60. CMB 2011 (vol 54 pp. 316) Mazhouda, Kamel  The Saddle-Point Method and the Li Coefficients In this paper, we apply the saddle-point method in conjunction with the theory of the NÃ¶rlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function$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 61. 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 62. 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 63. 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 64. 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 65. 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 66. 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 67. CMB 2009 (vol 53 pp. 102) Khan, Rizwanur  Spacings Between Integers Having Typically Many Prime Factors We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for$\delta$arbitrarily small and positive, the nearest neighbor spacings between integers n with$|\omega(n) - log log n| < (log log n)^{\delta}$obey the Poisson distribution law. Category:11K99 68. CMB 2009 (vol 53 pp. 187) Ünver, Sinan  On the Local Unipotent Fundamental Group Scheme We prove a local, unipotent, analog of Kedlaya's theorem for the pro-p part of the fundamental group of integral affine schemes in characteristic p. Category:11G25 69. 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 70. 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 71. CMB 2009 (vol 53 pp. 95) Ghioca, Dragos  Towards the Full Mordell-Lang Conjecture for Drinfeld Modules Let$\phi$be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of$\mathbb{G_a^g}$. We show that the intersection of X with a finite rank$\phi$-submodule of$\mathbb{G_a^g}$is finite. Keywords:Drinfeld module, Mordell-Lang conjectureCategories:11G09, 11G10 72. CMB 2009 (vol 53 pp. 140) Mukunda, Keshav  Pisot Numbers from$\{ 0, 1 \}$-Polynomials A \emph{Pisot number} is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a \emph{Salem number} is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial Â- one with$\{0,1\}$-coefficients Â- and shows that they form a strictly increasing sequence with limit$(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials. Categories:11R06, 11R09, 11C08 73. CMB 2009 (vol 53 pp. 58) Dąbrowski, Andrzej; Jędrzejak, Tomasz  Ranks in Families of Jacobian Varieties of Twisted Fermat Curves In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series. Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical heightCategories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15 74. CMB 2009 (vol 52 pp. 511) Bonciocat, Anca Iuliana; Bonciocat, Nicolae Ciprian  The Irreducibility of Polynomials That Have One Large Coefficient and Take a Prime Value We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value. Keywords:Estimates for polynomial roots, irreducible polynomialsCategories:11C08, 11R09 75. CMB 2009 (vol 52 pp. 583) Konstantinou, Elisavet; Kontogeorgis, Aristides  Computing Polynomials of the Ramanujan$t_n$Class Invariants We compute the minimal polynomials of the Ramanujan values$t_n$, where$n\equiv 11 \mod 24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field$\mathbb{Q}(\sqrt{-n})\$ and have much smaller coefficients than the Hilbert polynomials. Categories:11R29, 33E05, 11R20
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