1. CMB Online first
 Lee, Hao

Irregular Weight one points with $D_{4}$ Image
Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.
Keywords:weight one points, irregular, dihedral image, generalized eigenform, eigencurve, tangent space, Categories:11F33, 11F80 

2. CMB 2011 (vol 56 pp. 39)
 Ben Amara, Jamel

Comparison Theorem for Conjugate Points of a Fourthorder Linear Differential Equation
In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systemsconjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.
Keywords:fourthorder linear differential equation, conjugate points, systemconjugate points, subwronskians Categories:47E05, 34B05, 34C10 

3. CMB 2011 (vol 56 pp. 500)
 Browning, T. D.

The LangWeil 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 

4. CMB 2011 (vol 55 pp. 842)
5. 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 curves Category:11G05 

6. CMB 2009 (vol 53 pp. 340)
7. CMB 2004 (vol 47 pp. 398)
 McKinnon, David

A Reduction of the BatyrevManin Conjecture for Kummer Surfaces
Let $V$ be a $K3$ surface defined over a number field $k$. The
BatyrevManin conjecture for $V$ states that for every nonempty open
subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating
rational curves such that the density of rational points on $UZ_U$ is
strictly less than the density of rational points on $Z_U$. Thus,
the set of rational points of $V$ conjecturally admits a stratification
corresponding to the sets $Z_U$ for successively smaller sets $U$.
In this paper, in the case that $V$ is a Kummer surface, we prove that
the BatyrevManin conjecture for $V$ can be reduced to the
BatyrevManin conjecture for $V$ modulo the endomorphisms of $V$
induced by multiplication by $m$ on the associated abelian surface
$A$. As an application, we use this to show that given some restrictions
on $A$, the set of rational points of $V$ which lie on rational curves
whose preimages have geometric genus 2 admits a stratification of
Keywords:rational points, BatyrevManin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 

8. CMB 2003 (vol 46 pp. 373)
 Laugesen, Richard S.; Pritsker, Igor E.

Potential Theory of the FarthestPoint Distance Function
We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 

9. CMB 2002 (vol 45 pp. 337)
 Chen, Imin

Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes
For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, nonsplit Cartan Categories:11G05, 11F80 

10. CMB 2000 (vol 43 pp. 294)
11. CMB 1999 (vol 42 pp. 118)
 Rao, T. S. S. R. K.

Points of Weak$^\ast$Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$
For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weaknorm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the RadonNikodym property.
Keywords:Points of weak$^\ast$norm continuity, space of vector valued weakly continuous functions, $M$ideals Categories:46B20, 46E40 

12. CMB 1997 (vol 40 pp. 356)
 Mazet, Pierre

Principe du maximum et lemme de Schwarz, a valeurs vectorielles
Nous {\'e}tablissons un
th{\'e}or{\`e}me pour les fonctions holomorphes {\`a} valeurs dans une
partie convexe ferm{\'e}e. Ce th{\'e}or{\`e}me pr{\'e}cise
la position des coefficients de Taylor de telles fonctions et peut
{\^e}tre consid{\'e}r{\'e} comme une g{\'e}n{\'e}ralisation des
in{\'e}galit{\'e}s de Cauchy. Nous montrons alors comment ce
th{\'e}or{\`e}me permet de retrouver des versions connues du principe
du maximum et d'obtenir de nouveaux r{\'e}sultats sur les
applications holomorphes {\`a} valeurs vectorielles.
Keywords:Principe du maximum, lemme de Schwarz, points extr{Ã©maux. Categories:30C80, 32A30, 46G20, 52A07 
