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76. CMB 2006 (vol 49 pp. 296)

Sch"utt, Matthias
 On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three non-rigid Calabi--Yau 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 two-dimensional 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 two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27. Categories:14J32, 11F11, 11F23, 20C12

77. CMB 2006 (vol 49 pp. 196)

 Another Proof of Totaro's Theorem on $E_8$-Torsors We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$. Categories:11E72, 14M17, 20G15

78. CMB 2006 (vol 49 pp. 72)

Dwilewicz, Roman J.
 Additive Riemann--Hilbert 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 Cauchy-Riemann 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 bundlesCategories:32F20, 14F05, 32C16

79. CMB 2006 (vol 49 pp. 11)

Bevelacqua, Anthony J.; Motley, Mark J.
 Going-Down 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 ConjectureCategories:12F, 14G

80. 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 Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ 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 $\C^{n-1}$, we solve this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$ as in the title. Keywords:variables, Abhyankar--Sathaye Embedding ProblemCategories:14R10, 14R25

81. CMB 2005 (vol 48 pp. 547)

Fehér, L. M.; Némethi, A.; Rimányi, R.
 Degeneracy of 2-Forms and 3-Forms We study some global aspects of differential complex 2-forms and 3-forms 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, 2-forms, 3-forms, Thom polynomials, global singularity theoryCategories:14N10, 57R45

82. CMB 2005 (vol 48 pp. 428)

Miyamoto, Roland; Top, Jaap
 Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) and fibre type for elliptic curves over discrete valued fields of equal characteristic~3. Along the same lines, partial results are obtained in equal characteristic~2. Categories:14H52, 14K15, 11G07, 11G05, 12J10

83. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
 Logarithms and the Topology of the Complement of a Hypersurface This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$. Keywords:Logarithm, homology groups and periodsCategories:32Q55, 14F45

84. 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 algebraCategories:14M25, 52B20

85. CMB 2005 (vol 48 pp. 180)

Cynk, Sławomir; Meyer, Christian
 Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds We study Calabi--Yau 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 Calabi--Yau manifolds and 14 families with $h^{1,2}=1$. The modularity conjecture is verified for all the rigid examples. Keywords:Calabi--Yau, double coverings, modular formsCategories:14G10, 14J32

86. 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 self-product 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 non-torsion cycle in this group. Categories:14C25, 19D45

87. CMB 2005 (vol 48 pp. 203)

de Quehen, Victoria E.; Roberts, Leslie G.
 Non-Cohen--Macaulay Projective Monomial Curves with Positive ${h}$-Vector We find an infinite family of projective monomial curves all of which have $h$-vector with no negative values and are not Cohen-Macaulay. Category:14H45

88. CMB 2005 (vol 48 pp. 90)

Jeffrey, Lisa C.; Mare, Augustin-Liviu
 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{Je-We}. Categories:14D20, 14P05

89. CMB 2004 (vol 47 pp. 566)

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

90. CMB 2004 (vol 47 pp. 398)

McKinnon, David
 A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin 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 $U-Z_U$ 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 Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin 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, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, heightCategories:11G35, 14G05

91. CMB 2004 (vol 47 pp. 264)

McKinnon, David
 Counting Rational Points on Ruled Varieties In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$. Categories:11G50, 11D45, 11D04, 14G05

92. CMB 2004 (vol 47 pp. 271)

Naumann, Niko
 Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$. Categories:11G10, 14K15

93. CMB 2004 (vol 47 pp. 22)

Goto, Yasuhiro
 A Note on the Height of the Formal Brauer Group of a $K3$ Surface Using weighted Delsarte surfaces, we give examples of $K3$ surfaces in positive characteristic whose formal Brauer groups have height equal to $5$, $8$ or $9$. These are among the four values of the height left open in the article of Yui \cite{Y}. Keywords:formal Brauer groups, $K3$ surfaces in positive, characteristic, weighted Delsarte surfacesCategories:14L05, 14J28

94. CMB 2003 (vol 46 pp. 546)

Long, Ling
 $L$-Series of Certain Elliptic Surfaces In this paper, we study the modularity of certain elliptic surfaces by determining their $L$-series through their monodromy groups. Categories:14J27, 11M06

95. CMB 2003 (vol 46 pp. 575)

Marshall, M.
 Optimization of Polynomial Functions This paper develops a refinement of Lasserre's algorithm for optimizing a polynomial on a basic closed semialgebraic set via semidefinite programming and addresses an open question concerning the duality gap. It is shown that, under certain natural stability assumptions, the problem of optimization on a basic closed set reduces to the compact case. Categories:14P10, 46L05, 90C22

96. CMB 2003 (vol 46 pp. 495)

Baragar, Arthur
 Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two Let $V$ be an algebraic K3 surface defined over a number field $K$. Suppose $V$ has Picard number two and an infinite group of automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we introduce the notion of a vector height $\mathbf{h} \colon V \to \Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical vector height $\widehat{\mathbf{h}}$ with the following properties: \begin{gather*} \widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\ h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1), \end{gather*} where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of $\sigma$ (the pullback of $\sigma^{-1}$), and $h_D$ is a Weil height associated to the divisor $D$. The bounded function implied by the $O(1)$ does not depend on $P$. This allows us to attack some arithmetic problems. For example, we show that the number of rational points with bounded logarithmic height in an $\mathcal{A}$-orbit satisfies $$N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q) Categories:11G50, 14J28, 14G40, 14J50, 14G05 97. CMB 2003 (vol 46 pp. 321) Ballico, E.  Discreteness For the Set of Complex Structures On a Real Variety Let X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X\times Y \cong X\times W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X\times X^\sigma \cong W\times W^\sigma (where A^\sigma denotes the complex conjugate of any variety A) is at most countable. Categories:32J18, 14J99, 14P99 98. CMB 2003 (vol 46 pp. 429) Sastry, Pramathanath; Tong, Yue Lin L.  The Grothendieck Trace and the de Rham Integral On a smooth n-dimensional complete variety X over {\mathbb C} we show that the trace map {\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C} arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with$$ (2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C}  under the Dolbeault isomorphism. Categories:14F10, 32A25, 14A15, 14F05, 18E30

99. CMB 2003 (vol 46 pp. 400)

Marshall, M.
 Approximating Positive Polynomials Using Sums of Squares The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways. Categories:14P10, 44A60

100. CMB 2003 (vol 46 pp. 323)

Chamberland, Marc
 Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2 \rightarrow \mathbb{R}^2$ whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or $F$ is a polynomial. Specifically, $F=(u,v)$ must take the form \begin{gather*} u = ax + by + \beta \phi(\alpha x + \beta y) + e \\ v = cx + dy - \alpha \phi(\alpha x + \beta y) + f \end{gather*} for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and a $C^1$ function $\phi$ in one variable. If, in addition, the function $\phi$ is not affine, then $$\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.$$ This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge--Amp\`ere equation. Keywords:Jacobian Conjecture, injectivity, Monge--AmpÃ¨re equationCategories:26B10, 14R15, 35L70
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