scholarly journals ABELIAN -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

2017 ◽  
Vol 5 ◽  
Author(s):  
ANTHONY VÁRILLY-ALVARADO ◽  
BIANCA VIRAY
Keyword(s):  
Elliptic Curves ◽  
Abelian Surface ◽  
Uniform Bound ◽  
Brauer Groups ◽  
Fixed Degree ◽  
Related Family ◽  
The Family ◽  
Kummer Surfaces ◽  
Principal Homogeneous Space ◽  
Group Modulo

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\,Y/\text{Br}_{1}\,Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd torsion of $\text{Br}Y/\text{Br}_{1}Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $\ell$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Néron–Severi lattice, $\#(\text{Br}Y/\text{Br}_{1}Y)[\ell ^{\infty }]$ is bounded by a constant that depends only on $\ell$, $r$, and the discriminant.

Family Justice, “Asian Values” and the Politics of Ambivalence in Singapore

2018 ◽  
Vol 46 (4-5) ◽  
pp. 467-483
Author(s):  
Daniel P.S. Goh
Keyword(s):  
Nuclear Family ◽  
Family Values ◽  
Policy Changes ◽  
Asian Values ◽  
Related Family ◽  
New Family ◽  
Rights Of The Child ◽  
The Family ◽  
Family Types ◽  
Discursive Field

Abstract In recent years, Singapore made significant reforms towards the establishment of a dedicated family justice system, setting up the Family Justice Courts and enacting new laws to better manage the divorce process and the protection of children. Related policy changes have also been implemented to provide and support families that were previously considered non-traditional and even deviant. Rhetorically, the state, led by the long-ruling People’s Action Party, continues to champion the modern nuclear family with heterosexual marriage at its core as the normal “traditional” form of the family and the bedrock of conservative “Asian values” defining society and politics in Singapore. However, what the judiciary espouse as the new family justice paradigm and the related family justice practices, together with the shifts in social policy towards different family types, are changing the texture of the dominant conservatism rallied by “Asian values” discourse. This article locates and analyses the incipient paradigm shift in the rising pluralism of family forms and the influence of international legal developments in protecting the rights of the child and interventionist family law. By attempting to bridge the Weberian chasm of doing sociology as a vocation and doing politics as a vocation (as an opposition Member of Parliament), I show that the family justice paradigm has opened up the discursive field on the family and produce the politics of ambivalence caught between family justice and Asian family values. I argue for a relational family justice paradigm as a way to move beyond the politics of ambivalence.

scholarly journals Moments of the Critical Values of Families of Elliptic Curves, with Applications

2010 ◽  
Vol 62 (5) ◽  
pp. 1155-1181 ◽  
Author(s):  
Matthew P. Young
Keyword(s):  
Elliptic Curves ◽  
Large Family ◽  
Critical Values ◽  
Central Values ◽  
The Family ◽  
Order Of Magnitude ◽  
Rank 2 ◽  
Family Based ◽  
First Moment ◽  
Positive Rank

AbstractWe make conjectures on the moments of the central values of the family of all elliptic curves and on themoments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family.Furthermore, as arithmetical applications, we make a conjecture on the distribution of ap's amongst all rank 2 elliptic curves and show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet 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 ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU 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 Batyrev-Manin type.

scholarly journals Genus-2 curves and Jacobians with a given number of points

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Polynomial Time ◽  
Rational Points ◽  
The Other ◽  
Base Field ◽  
Arbitrary Base ◽  
The Family ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

scholarly journals 2-Descent on elliptic curves and rational points on certain Kummer surfaces

2005 ◽  
Vol 198 (2) ◽  
pp. 448-483 ◽  
Author(s):  
Alexei Skorobogatov ◽  
Peter Swinnerton-Dyer
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Kummer Surfaces

The Convergence of Some Products in the Adams Spectral Sequence

2015 ◽  
Vol 117 (2) ◽  
pp. 304
Author(s):  
Yuyu Wang ◽  
Jianbo Wang
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
May Spectral Sequence ◽  
Related Family ◽  
The Family

In this paper, we will use the family of homotopy elements $\zeta_n\in\pi_*S$, represented by $h_0b_n\in \operatorname{Ext}_A^{3,p^{n+1} q+q}(\mathsf{Z}_p, \mathsf{Z}_p)$ in the Adams spectral sequence, to detect a $\zeta_n$-related family $\gamma_{s+3}\beta_2\zeta_{n-1}$ in $\pi_*S$. Our main methods are the Adams spectral sequence and the May spectral sequence, here prime $p\geq 7$, $n>3$, $q=2(p-1)$.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

2006 ◽  
Vol 02 (04) ◽  
pp. 569-590 ◽  
Author(s):  
BORKA JADRIJEVIĆ ◽  
VOLKER ZIEGLER
Keyword(s):  
Number Field ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Root Of Unity ◽  
Related Family ◽  
The Family ◽  
Thue Equation ◽  
Imaginary Quadratic Number ◽  
Pellian Equations

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

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