51. CMB 2007 (vol 50 pp. 196)
 Fernández, Julio; González, Josep; Lario, JoanC.

Plane Quartic Twists of $X(5,3)$
Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genusthree case $N=5$.
Categories:11F03, 11F80, 14G05 

52. CMB 2007 (vol 50 pp. 243)
 Langlands, Robert P.

Un nouveau point de repÃ¨re dans la thÃ©orie des formes automorphes
Dans le papier Beyond Endoscopy une id\'ee pour entamer la
fonctorialit\'e en utilisant la formule des traces a \'et\'e
introduite. Maints probl\`emes, l'existence d'une limite convenable
de la formule des traces, est eqquiss\'ee dans cette note
informelle mais seulement pour $GL(2)$ et les corps des fonctions
rationelles sur un corps fini et en ne pas resolvant
bon nombre de questions.
Categories:32N10, 14xx 

53. CMB 2007 (vol 50 pp. 215)
 Kloosterman, Remke

Elliptic $K3$ Surfaces with Geometric MordellWeil Rank $15$
We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric MordellWeil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$surfaces with geometric MordellWeil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 

54. CMB 2007 (vol 50 pp. 161)
55. CMB 2007 (vol 50 pp. 126)
 Ongay, Fausto

$\varphi$Dialgebras and a Class of Matrix ``Coquecigrues''
Starting with the Leibniz algebra defined by a $\varphi$dialgebra, we
construct examples of ``coquecigrues,'' in the sense of Loday, that is to
say, manifolds whose tangent structure at a distinguished point coincides
with that of the Leibniz algebra. We discuss some possible
implications and generalizations of this construction.
Keywords:Leibniz algebras, dialgebras Category:14M30 

56. CMB 2007 (vol 50 pp. 105)
 Klep, Igor

On Valuations, Places and Graded Rings Associated to $*$Orderings
We study natural $*$valuations, $*$places and graded $*$rings
associated with $*$ordered rings.
We prove that the natural $*$valuation is always quasiOre and is
even quasicommutative (\emph{i.e.,} the corresponding graded $*$ring is
commutative), provided the ring contains an imaginary unit.
Furthermore, it is proved that the graded $*$ring is isomorphic
to a twisted semigroup algebra. Our results are applied to answer a question
of Cimpri\v c regarding $*$orderability of quantum
groups.
Keywords:$*$orderings, valuations, rings with involution Categories:14P10, 16S30, 16W10 

57. CMB 2006 (vol 49 pp. 560)
 Luijk, Ronald van

A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues
In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

58. CMB 2006 (vol 49 pp. 592)
 Sarti, Alessandra

Group Actions, Cyclic Coverings and Families of K3Surfaces
In this paper we describe six pencils of $K3$surfaces which have
large Picard number ($\rho=19,20$) and each contains precisely five
special fibers: four have ADE singularities and one is
nonreduced. In particular, we characterize these surfaces as cyclic
coverings of some $K3$surfaces described in a recent paper by Barth
and the author.
In many cases, using
3divisible sets, resp., 2divisible sets, of rational curves and
lattice theory, we describe explicitly the Picard lattices.
Categories:14J28, 14L30, 14E20, 14C22 

59. CMB 2006 (vol 49 pp. 464)
 Ravindra, G. V.

A Note on Detecting Algebraic Cycles
The purpose of this note is to show that the homologically trivial
cycles contructed by Clemens and their generalisations
due to Paranjape can be detected by the technique of
spreading out. More precisely, we associate to these cycles (and the
ambient varieties in which they live) certain families which arise
naturally and which are defined over $\bbC$ and show that these
cycles, along with their relations, can be detected in the singular
cohomology of the total space of these families.
Category:14C25 

60. CMB 2006 (vol 49 pp. 296)
 Sch"utt, Matthias

On the Modularity of Three CalabiYau Threefolds With Bad Reduction at 11
This paper investigates the modularity of three
nonrigid CalabiYau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$adic cohomology groups are shown to split into
twodimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
twodimensional 2adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hessepencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 

61. CMB 2006 (vol 49 pp. 270)
62. CMB 2006 (vol 49 pp. 196)
63. CMB 2006 (vol 49 pp. 11)
 Bevelacqua, Anthony J.; Motley, Mark J.

GoingDown Results for $C_{i}$Fields
We search for theorems that, given a $C_i$field $K$ and a subfield $k$ of $K$, allow
us to conclude that $k$ is a $C_j$field for some $j$. We give appropriate theorems in
the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$
is an algebraic extension. Here we are able to prove some results, and make conjectures. We
also point out the connection between these questions and Lang's conjecture on nonreal function
fields over a real closed field.
Keywords:$C_i$fields, Lang's Conjecture Categories:12F, 14G 

64. CMB 2006 (vol 49 pp. 72)
 Dwilewicz, Roman J.

Additive RiemannHilbert Problem in Line Bundles Over $\mathbb{CP}^1$
In this note we consider $\overline\partial$problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any CauchyRiemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.
Keywords:$\overline\partial$problem, cohomology groups, line bundles Categories:32F20, 14F05, 32C16 

65. CMB 2005 (vol 48 pp. 622)
 Vénéreau, Stéphane

Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables
The AbhyankarSathaye Embedded Hyperplane Problem asks whe\ther any
hypersurface of $\C^n$ isomorphic to $\C^{n1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.
Keywords:variables, AbhyankarSathaye Embedding Problem Categories:14R10, 14R25 

66. CMB 2005 (vol 48 pp. 547)
 Fehér, L. M.; Némethi, A.; Rimányi, R.

Degeneracy of 2Forms and 3Forms
We study some global aspects of differential complex 2forms and 3forms
on complex manifolds.
We compute the cohomology classes represented by the sets of points
on a manifold where such a form degenerates in various senses,
together with other similar cohomological obstructions.
Based on these results and a formula for projective
representations, we calculate the degree of the projectivization
of certain orbits of the representation $\Lambda^k\C^n$.
Keywords:Classes of degeneracy loci, 2forms, 3forms, Thom polynomials, global singularity theory Categories:14N10, 57R45 

67. CMB 2005 (vol 48 pp. 428)
68. CMB 2005 (vol 48 pp. 473)
69. CMB 2005 (vol 48 pp. 414)
 Kaveh, Kiumars

Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex
projective variety with a holomorphic vector field with isolated
zero set $Z$. From the results of Carrell and Lieberman
there exists a filtration
$F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of
$\c$valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X,
\c)$ as graded algebras. In this note, for a smooth projective
toric variety and a vector field generated by the action of a
$1$parameter subgroup of the torus, we work out this filtration.
Our main result is an explicit connection between this filtration
and the polytope algebra of $X$.
Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra Categories:14M25, 52B20 

70. CMB 2005 (vol 48 pp. 180)
 Cynk, Sławomir; Meyer, Christian

Geometry and Arithmetic of Certain Double Octic CalabiYau Manifolds
We study CalabiYau manifolds constructed as double coverings of
$\mathbb{P}^3$ branched along an octic surface. We give a list of 87
examples corresponding to arrangements of eight planes defined over
$\mathbb{Q}$. The Hodge numbers are computed for all examples. There are
10 rigid CalabiYau manifolds and 14 families with $h^{1,2}=1$. The
modularity conjecture is verified for all the rigid examples.
Keywords:CalabiYau, double coverings, modular forms Categories:14G10, 14J32 

71. CMB 2005 (vol 48 pp. 237)
 Kimura, Kenichiro

Indecomposable Higher Chow Cycles
Let $X$ be a projective smooth variety over a field $k$.
In the first part we show that
an indecomposable element in $CH^2(X,1)$ can be lifted
to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function
field of 1 variable over $k$. We also show that if $X$ is the selfproduct
of an elliptic curve over $\Q$ then the $\Q$vector space of
indecomposable cycles
$CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional.
In the second part we give a new
definition of the group of indecomposable cycles
of $CH^3(X,2)$ and give an example of nontorsion
cycle in this group.
Categories:14C25, 19D45 

72. CMB 2005 (vol 48 pp. 203)
73. CMB 2005 (vol 48 pp. 90)
 Jeffrey, Lisa C.; Mare, AugustinLiviu

Products of Conjugacy Classes in $SU(2)$
We obtain a complete description of the conjugacy classes
$C_1,\dots,C_n$ in $SU(2)$ with the property that $C_1\cdots
C_n=SU(2)$. The basic instrument is a characterization of the
conjugacy classes $C_1,\dots,C_{n+1}$ in $SU(2)$ with $C_1\cdots
C_{n+1}\ni I$, which generalizes a result of \cite{JeWe}.
Categories:14D20, 14P05 

74. CMB 2004 (vol 47 pp. 566)
 Koike, Kenji

Algebraicity of some Weil Hodge Classes
We show that the Prym map for 4th cyclic \'etale covers of curves
of genus 4 is a dominant morphism to a Shimura variety for a family
of Abelian 6folds of Weil type. According to the result of Schoen,
this implies algebraicity of Weil classes for this family.
Category:14C30 

75. 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 
