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1. CMB Online first

Ide, Joshua; Jones, Lenny
Infinite Families of $A_4$-Sextic Polynomials
In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group $A_4$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Keywords:Galois group, sextic polynomial, inverse Galois theory, irreducible polynomial
Categories:12F10, 12F12, 11R32, 11R09

2. CMB Online first

Franc, Cameron; Mason, Geoffrey
Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$
We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued modular form on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

Keywords:vector-valued modular form, modular group, bounded denominators
Categories:11F41, 11G99

3. CMB Online first

Fujita, Yasutsugu; Miyazaki, Takafumi
Jeśmanowicz' Conjecture with Congruence Relations. II
Let $a,b$ and $c$ be primitive Pythagorean numbers such that $a^{2}+b^{2}=c^{2}$ with $b$ even. In this paper, we show that if $b_0 \equiv \epsilon \pmod{a}$ with $\epsilon \in \{\pm1\}$ for certain positive divisors $b_0$ of $b$, then the Diophantine equation $a^{x}+b^{y}=c^z$ has only the positive solution $(x,y,z)=(2,2,2)$.

Keywords:exponential Diophantine equations, Pythagorean triples, Pell equations
Categories:11D61, 11D09

4. CMB Online first

Kane, Daniel M.; Kominers, Scott Duke
Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathop{\textrm{lcm}}(u_0,u_1,\dots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ that improve upon those obtained previously when either $u_0$ or $n$ is large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for $u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.

Keywords:least common multiple, arithmetic progression

5. CMB Online first

Lei, Antonio
Factorisation of Two-variable $p$-adic $L$-functions
Let $f$ be a modular form which is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, he showed that, as in the one-variable case, Pollack's plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where $a_p\ne0$ using Sprung's logarithmic matrix.

Keywords:modular forms, p-adic L-functions, supersingular primes
Categories:11S40, 11S80

6. CMB Online first

Schoen, Tomasz
On Convolutions of Convex Sets and Related Problems
We prove some results concerning covolutions, the additive energy and sumsets of convex sets and its generalizations. In particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq \mathbb R$ has the property that for every fixed $1\leqslant d\lt n,$ all differences $a_i-a_{i-d}$, $d\lt i\lt n,$ are distinct, then $|A+A|\gg |A|^{3/2+c}$ for a constant $c\gt 0.$

Keywords:convex sets, additive energy, sumsets

7. CMB 2013 (vol 57 pp. 381)

Łydka, Adrian
On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve
We study analytic properties function $m(z, E)$, which is defined on the upper half-plane as an integral from the shifted $L$-function of an elliptic curve. We show that $m(z, E)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m(z, E)$ in the strip $|\Im{z}|\lt 2\pi$.

Keywords:L-function, Möbius function, explicit formulae, elliptic curve
Categories:11M36, 11G40

8. CMB 2013 (vol 56 pp. 827)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
Erratum to ``Quantum Limits of Eisenstein Series and Scattering States''
This paper provides an erratum to Y. N. Petridis, N. Raulf, and M. S. Risager, ``Quantum Limits of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published online 2012-02-03,

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 8G25, 35P25

9. CMB 2013 (vol 56 pp. 673)

Ayadi, K.; Hbaib, M.; Mahjoub, F.
Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic
In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type \begin{equation} \alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}} \end{equation} where $(A, B, C)\in (\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$. In particular, we will give, under some conditions on the polynomials $A$, $B$ and $C$, well approximated elements satisfying this equation.

Keywords:diophantine approximation, formal power series, continued fraction
Categories:11J61, 11J70

10. CMB 2012 (vol 56 pp. 570)

Hoang, Giabao; Ressler, Wendell
Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups
In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in any Hecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.

Keywords:Hecke groups, conjugacy class, quadratic forms
Categories:11F06, 11E16, 11A55

11. CMB 2012 (vol 57 pp. 105)

Luca, Florian; Shparlinski, Igor E.
On the Counting Function of Elliptic Carmichael Numbers
We give an upper bound for the number elliptic Carmichael numbers $n \le x$ that have recently been introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss several possible ways for further improvements.

Keywords:elliptic Carmichael numbers, applications of sieve methods
Categories:11Y11, 11N36

12. CMB 2012 (vol 56 pp. 695)

Banks, William D.; Güloğlu, Ahmet M.; Yeager, Aaron M.
Carmichael meets Chebotarev
For any finite Galois extension $K$ of $\mathbb Q$ and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$, we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is $C$. This result implies that for every natural number $n$ there are infinitely many Carmichael numbers of the form $a^2+nb^2$ with $a,b\in\mathbb Z $.

Keywords:Carmichael numbers, Chebotarev density theorem
Categories:11N25, 11R45

13. CMB 2012 (vol 56 pp. 785)

Liu, Zhixin
Small Prime Solutions to Cubic Diophantine Equations
Let $a_1, \cdots, a_9$ be non-zero integers and $n$ any integer. Suppose that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$ for $1 \leq i \lt j \leq 9$. In this paper we prove that (i) if $a_j$ are not all of the same sign, then the above cubic equation has prime solutions satisfying $p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{14+\varepsilon};$ and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{43+\varepsilon}$, then the cubic equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$. This result is the extension of the linear and quadratic relative problems.

Keywords:small prime, Waring-Goldbach problem, circle method
Categories:11P32, 11P05, 11P55

14. CMB 2012 (vol 56 pp. 759)

Issa, Zahraa; Lalín, Matilde
A Generalization of a Theorem of Boyd and Lawton
The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).

Keywords:Mahler measure, polynomial
Categories:11R06, 11R09

15. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
On the Average Number of Square-Free Values of Polynomials
We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture.

Keywords:polynomials, square-free numbers

16. CMB 2012 (vol 56 pp. 829)

Pollack, Paul
On Mertens' Theorem for Beurling Primes
Let $1 \lt p_1 \leq p_2 \leq p_3 \leq \dots$ be an infinite sequence $\mathcal{P}$ of real numbers for which $p_i \to \infty$, and associate to this sequence the \emph{Beurling zeta function} $\zeta_{\mathcal{P}}(s):= \prod_{i=1}^{\infty}(1-p_i^{-s})^{-1}$. Suppose that for some constant $A\gt 0$, we have $\zeta_{\mathcal{P}}(s) \sim A/(s-1)$, as $s\downarrow 1$. We prove that $\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens: $\prod_{p_i \leq x}(1-1/p_i)^{-1} \sim A \e^{\gamma} \log{x}$, as $x\to\infty$. Here $\e = 2.71828\ldots$ is the base of the natural logarithm and $\gamma = 0.57721\ldots$ is the usual Euler--Mascheroni constant. This strengthens a recent theorem of Olofsson.

Keywords:Beurling prime, Mertens' theorem, generalized prime, arithmetic semigroup, abstract analytic number theory
Categories:11N80, 11N05, 11M45

17. CMB 2012 (vol 56 pp. 602)

Louboutin, Stéphane R.
Resultants of Chebyshev Polynomials: A Short Proof
We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes and V. Trevisan.

Keywords:resultant, Chebyshev polynomials, cyclotomic polynomials
Categories:11R09, 11R04

18. CMB 2012 (vol 56 pp. 814)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
Quantum Limits of Eisenstein Series and Scattering States
We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

Keywords:quantum limits, Eisenstein series, scattering poles
Categories:11F72, 58G25, 35P25

19. CMB 2012 (vol 56 pp. 723)

Bérczes, Attila; Luca, Florian
On the Sum of Digits of Numerators of Bernoulli Numbers
Let $b\gt 1$ be an integer. We prove that for almost all $n$, the sum of the digits in base $b$ of the numerator of the Bernoulli number $B_{2n}$ exceeds $c\log n$, where $c:=c(b)\gt 0$ is some constant depending on $b$.

Keywords:Bernoulli numbers, sums of digits

20. CMB 2012 (vol 56 pp. 520)

Elbasraoui, Abdelkrim; Sebbar, Abdellah
Equivariant Forms: Structure and Geometry
In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\operatorname{SL}_2(\mathbb{Z})$ by means of the cross-ratio, the weight 2 modular forms, the quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

Keywords:equivariant forms, modular forms, Schwarz derivative, cross-ratio, differential forms

21. CMB 2012 (vol 56 pp. 544)

Gauthier, P. M.
Universally Overconvergent Power Series via the Riemann Zeta-function
The Riemann zeta-function is employed to generate universally overconvergent power series.

Keywords:overconvergence, zeta-function
Categories:30K05, 11M06

22. CMB 2011 (vol 56 pp. 225)

Agashe, Amod
On the Notion of Visibility of Torsors
Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$. Let $x$ be an element of $H^1(\mathbf{Q},A)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we clarify the relation between the two definitions.

Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group
Categories:11G35, 14G25

23. CMB 2011 (vol 56 pp. 412)

Sanders, T.
Structure in Sets with Logarithmic Doubling
Suppose that $G$ is an abelian group, $A \subset G$ is finite with $|A+A| \leq K|A|$ and $\eta \in (0,1]$ is a parameter. Our main result is that there is a set $\mathcal{L}$ such that \begin{equation*} |A \cap \operatorname{Span}(\mathcal{L})| \geq K^{-O_\eta(1)}|A| \quad\text{and}\quad |\mathcal{L}| = O(K^\eta\log |A|). \end{equation*} We include an application of this result to a generalisation of the Roth--Meshulam theorem due to Liu and Spencer.

Keywords:Fourier analysis, Freiman's theorem, capset problem

24. CMB 2011 (vol 56 pp. 510)

Dubickas, Artūras
Linear Forms in Monic Integer Polynomials
We prove a necessary and sufficient condition on the list of nonzero integers $u_1,\dots,u_k$, $k \geq 2$, under which a monic polynomial $f \in \mathbb{Z}[x]$ is expressible by a linear form $u_1f_1+\dots+u_kf_k$ in monic polynomials $f_1,\dots,f_k \in \mathbb{Z}[x]$. This condition is independent of $f$. We also show that if this condition holds, then the monic polynomials $f_1,\dots,f_k$ can be chosen to be irreducible in $\mathbb{Z}[x]$.

Keywords:irreducible polynomial, height, linear form in polynomials, Eisenstein's criterion
Categories:11R09, 11C08, 11B83

25. CMB 2011 (vol 56 pp. 500)

Browning, T. D.
The Lang--Weil Estimate for Cubic Hypersurfaces
An improved estimate is provided for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible, projective, cubic hypersurface that is not equal to a cone.

Keywords:cubic hypersurface, rational points, finite fields
Categories:11G25, 14G15
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