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Search: MSC category 14G ( Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] )

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1. 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 groupCategories:11G35, 14G25

2. CMB Online first

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 fieldsCategories:11G25, 14G15

3. CMB 2011 (vol 55 pp. 842)

Sairaiji, Fumio; Yamauchi, Takuya
 The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell-Weil rank over the maximal abelian extension $K^{\operatorname{ab}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C(K^{\operatorname{ab}})=\infty$ and for any abelian variety of $\operatorname{GL}_2$-type with trivial character. Keywords:Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian pointsCategories:11G05, 11D25, 14G25, 14K07

4. CMB 2009 (vol 53 pp. 58)

Dąbrowski, Andrzej; Jędrzejak, Tomasz
 Ranks in Families of Jacobian Varieties of Twisted Fermat Curves In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series. Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical heightCategories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15

5. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
 On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$ Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points. Categories:11G30, 14G05, 14G25

6. CMB 2007 (vol 50 pp. 486)

Cynk, S.; Hulek, K.
 Higher-Dimensional Modular\\Calabi--Yau Manifolds We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity. Categories:14G10, 14J32, 11G40

7. CMB 2007 (vol 50 pp. 196)

Fernández, Julio; González, Josep; Lario, Joan-C.
 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 genus-three case $N=5$. Categories:11F03, 11F80, 14G05

8. 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 non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it. Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theoryCategories:14G05, 14J28, 11D41

9. 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

10. 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

11. 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

12. 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

13. 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