Rational Equivalences on Products of Elliptic Curves in a Family

@article{Love2020RationalEO,
  title={Rational Equivalences on Products of Elliptic Curves in a Family},
  author={Jonathan Love},
  journal={arXiv: Algebraic Geometry},
  year={2020}
}
Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a… Expand

Figures and Tables from this paper

References

SHOWING 1-10 OF 28 REFERENCES
Zero Cycles on a Product of Elliptic Curves Over a p-adic Field
We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ with a combination of good or split multiplicative reduction. We assume that atExpand
Rational equivalence of 0-cycles on surfaces
We will consider in this note 0-cycles on a complete non-singular algebraic surface F over the field C o f complex numbers. We will use the language o f schemes, and every scheme will be assumedExpand
Advanced Topics in the Arithmetic of Elliptic Curves
In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rationalExpand
Root numbers and ranks in positive characteristic
Abstract For a global field K and an elliptic curve E η over K ( T ) , Silverman's specialization theorem implies rank ( E η ( K ( T ) ) ) ⩽ rank ( E t ( K ) ) for all but finitely many t ∈ P 1 ( K )Expand
Heights and the specialization map for families of abelian varieties.
Let C be a non-singular projective curve, and let A — > C be a (flat) family of abelian varieties, all defmed over a global field K. There are three natural height functions associated to such aExpand
Algebraic cycles and values of L-functions.
Let X be a smooth projective algebraic variety of dimension d over a number field k, and let n ̂ 0 be an integer, /-adic cohomology in degree /i, H(X^, Ot), is a representation space for Gal (Jc/k)Expand
On the rank of an elliptic surface
Abstract. Nagao has recently given a conjectural limit formula for the rank of an elliptic surface E in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture onExpand
INTERSECTION THEORY
I provide more details to the intersection theoretic results in [1]. CONTENTS 1. Transversality and tubular neighborhoods 1 2. The Poincaré dual of a submanifold 4 3. Smooth cycles and theirExpand
The Elliptic Curve Database for Conductors to 130000
TLDR
This paper reports on significant recent progress in enlarging the database of elliptic curves defined over ℚ to include all those of conductor N≤130000 and gives various statistics. Expand
Rank Distribution in a Family of Cubic Twists
In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank.Expand
...
1
2
3
...