1. CJM Online first
 Folha, Abigail; Penafiel, Carlos

Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$
In this article, we study complete surfaces $\Sigma$, isometrically
immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or
$\mathbb{S}^2\times\mathbb{R}$
having positive extrinsic curvature $K_e$. Let $K_i$ denote the
intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$
holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main
result of this article state that when such a surface is a topological
sphere it is rotational.
Keywords:Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surface Categories:53C42, 53C30 

2. CJM 2016 (vol 68 pp. 1120)
 Stange, Katherine E.

Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of nontorsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and nontorsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of nonsingular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called \emph{elliptic
troublemaker sequences}, which measure the failure of the NÃ©ron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
Keywords:elliptic divisibility sequence, Lang's conjecture, height functions Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52 

3. CJM 2016 (vol 68 pp. 625)
 Ingram, Patrick

Rigidity and Height Bounds for Certain Postcritically Finite Endomorphisms of $\mathbb P^N$
The morphism $f:\mathbb{P}^N\to\mathbb{P}^N$ is called postcritically finite
(PCF) if the forward image of the critical locus, under iteration
of $f$, has algebraic support. In the case $N=1$, a result of
Thurston implies that there are no algebraic families of PCF
morphisms, other than a wellunderstood exceptional class known
as the flexible LattÃ¨s maps. A related arithmetic result
states that the set of PCF morphisms corresponds to a set of
bounded height in the moduli space of univariate rational functions.
We prove corresponding results for a certain subclass of the
regular polynomial endomorphisms of $\mathbb{P}^N$, for any $N$.
Keywords:postcritically finite, arithmetic dynamics, heights Categories:37P15, 32H50, 37P30 

4. CJM 2015 (vol 67 pp. 1046)
 Dubickas, Arturas; Sha, Min; Shparlinski, Igor

Explicit Form of Cassels' $p$adic Embedding Theorem for Number Fields
In this paper, we mainly give a general explicit form of Cassels'
$p$adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.
Keywords:number field, $p$adic embedding, height, polynomial, cyclotomic field Categories:11R04, 11S85, 11G50, 11R09, 11R18 

5. CJM 2007 (vol 59 pp. 1284)
 Fukshansky, Lenny

On Effective Witt Decomposition and the CartanDieudonn{Ã© Theorem
Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a smallheight orthogonal
decomposition into onedimensional subspaces. Finally, we prove an
effective version of the CartanDieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.
Keywords:quadratic form, heights Categories:11E12, 15A63, 11G50 
