scholarly journals Efficient computation of rank of elliptic curves using hyperelliptic curves associated with the simplest cubic fields

2000 ◽  
Vol 76 (9) ◽  
pp. 129-131
Author(s):  
Jung Soo Kim
Keyword(s):  
Elliptic Curves ◽  
Hyperelliptic Curves ◽  
Efficient Computation ◽  
Cubic Fields

scholarly journals Torsion of rational elliptic curves over cubic fields

2016 ◽  
Vol 46 (6) ◽  
pp. 1899-1917 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Filip Najman ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

On a class of unirational varieties

1951 ◽  
Vol 47 (3) ◽  
pp. 496-503 ◽  
Author(s):  
L. Roth
Keyword(s):  
General Theory ◽  
Elliptic Curves ◽  
Hyperelliptic Curves ◽  
Considerable Part ◽  
First Case

A theorem of Castelnuovo, which has played a considerable part in the general theory of surfaces, states that any surface which contains a net of elliptic curves is either rational or elliptic scrollar; more precisely, in the first case it is proved that the surface is unirational, and that its unirational representation is obtained by adjoining the irrationality on which depends the determination of one of its points, while the rest of the conclusion follows from Castelnuovo's theorem on the rationality of plane involutions. A somewhat similar result holds for surfaces which contain a net of hyperelliptic curves: thus, it is shown by Castelnuovo (loc. cit.) that, if the characteristic series of the net is not compounded of a g½ the surface is either rational or hyperelliptic scrollar.

Elliptic curves with good reduction everywhere over cubic fields

2015 ◽  
Vol 11 (04) ◽  
pp. 1149-1164 ◽  
Author(s):  
Nao Takeshi
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Good Reduction ◽  
Cubic Fields

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

scholarly journals The Selmer groups and the ambiguous ideal class groups of cubic fields

1996 ◽  
Vol 54 (2) ◽  
pp. 267-274
Author(s):  
Yen-Mei J. Chen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Selmer Groups ◽  
Ideal Class ◽  
Selmer Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Cubic Fields ◽  
Rational Isogeny

In this paper, we study a family of elliptic curves with CM by which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields. We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.

scholarly journals Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

2017 ◽  
Vol 4 (1) ◽  
pp. 7-15 ◽  
Author(s):  
Robert Xin Dong
Keyword(s):  
Elliptic Curves ◽  
Bergman Kernel ◽  
Complex Structure ◽  
Elliptic Functions ◽  
Hyperelliptic Curves ◽  
Asymptotic Formulas ◽  
Holomorphic Family ◽  
Structure Information ◽  
Taylor Expansions ◽  
Bergman Kernel Asymptotics

Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

scholarly journals Torsion of elliptic curves over cyclic cubic fields

2018 ◽  
Vol 88 (319) ◽  
pp. 2443-2459 ◽  
Author(s):  
Maarten Derickx ◽  
Filip Najman
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields ◽  
Cyclic Cubic Fields
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