This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in Pn−1. The main tool we use is a comparison between an easily obtained embedding into Pn2−1 and another map into Pn2−1 that factors through the Segre embedding Pn−1×Pn−1 → Pn2−1. The c...

We develop the plus/minus p-Selmer group theory and plus/minus padic L-function theory for an elliptic curve E with complex multiplication over an abelian extension F of the imaginary quadratic field K given by the complex multiplication of E when p is a prime inert over K/Q (i.e. supersingular). As a result, we prove that the characteristic ideal of the Pontryagin dual of the plus/minus p-Selm...

Heegner points on the modular curve X0(N), and their images on elliptic curve factors E of the Jacobian, enjoy many remarkable properties. These points are the moduli of level structures with endomorphisms by the ring of integers of an imaginary quadratic field K. Their traces to E(K) have height given by the first derivative at s = 1 of the L-function of E over K (cf. [GZ86]), and their l-divi...

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A obtained by Ribet’s level raising theorem. For certain imaginary quadratic fields K satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of E and A over K have different...

We rst develop a notion of quadratic form on a locally compact abelian group. Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over Fp. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a xed compact open maximal isotro...

The study of n-Selmer groups elliptic curves over number fields in recent past has led to the discovery some deep results arithmetic curves. Given two E 1 and 2 a field K, Mazur–Rubin have defined them be companion if for every quadratic character χ K are isomorphic. prime p, they given sufficient conditions p r -Selmer terms mod-p congruences between We discuss an analogue this Bloch–Kato modu...

Global root numbers have played an important role in the study of rational points on abelian varieties since the discovery of the conjecture of Birch and Swinnerton-Dyer. The aim of this paper is to throw some new light on this intriguing and still largely conjectural relationship. The simplest avatar of this phenomenon is the parity conjecture which asserts that for an abelian variety A over a...

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example ...

In this paper, we study a class of elliptic curves over Q with Qtorsion group Z2×Z2, and prove that the average order of the 2-Selmer groups is bounded.