Rational Equivalences on Products of Elliptic Curves in a Family

On Silverman's conjecture for a family of elliptic curves

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

On computing rational torsion on elliptic curves

We introduce an l-adic algorithm to efficiently determine the group of rational torsion points on an elliptic curve. We also make a conjecture about the discriminant of the m-division polynomial of an elliptic curve.

Principal Polarizations on Products of Elliptic Curves

An abelian variety admits only a finite number of isomorphism classes of principal polarizations. The paper gives an interpretation of this number in terms of class numbers of definite Hermitian forms in the case of a product of elliptic curves without complex multiplication. In the case of a self-product of an elliptic curve, as well as in the two-dimensional case, classical class number compu...

Products of CM elliptic curves

Massey and Fukaya Products on Elliptic Curves

This note is a continuation of [6]. Its goal is to show that some higher Massey products on elliptic curve can be computed as higher compositions in Fukaya category of the dual symplectic torus in accordance with the homological mirror conjecture of M. Kontsevich [4]. Namely, we consider triple Massey products of very simple type which are uniquely defined, compute them in terms of theta-functi...