in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.

For any abelian variety J over a global field k and an isogeny φ : J → J , the Selmer group Sel(J, k) is a subgroup of the Galois cohomology group H(Gal(k/k), J [φ]), defined in terms of local data. When J is the Jacobian of a cyclic cover of P of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is m...

A formula is given for the dimension of the Selmer group of the rational three-isogeny of elliptic curves of the form y2 = x3+a(x−b)2. The formula is in terms of the three-ranks of the quadratic number fields Q( √ a) and Q( √ −3a) and various aspects of the arithmetic of these number fields. In addition a duality theorem is used to relate the dimension of the Selmer group of the three-isogeny w...

In this paper we examine diierences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical diierences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention brieey ongoing research i n to generalizing such methods to t...

Let E be an elliptic curve over Q and A another elliptic curve over a real quadratic number field. We construct a Q-motive of rank 8, together with a distinguished class in the associated Bloch–Kato Selmer group, using Hirzebruch–Zagier cycles, that is, graphs of Hirzebruch–Zagier morphisms. We show that, under certain assumptions on E and A, the non-vanishing of the central critical value of t...

In this paper, we develop the idea in [16] to obtain finer results on the structure of Selmer modules for p-adic representations than the usual main conjecture in Iwasawa theory. We determine the higher Fitting ideals of the Selmer modules under several assumptions. Especially, we describe the structure of the classical Selmer group of an elliptic curve over Q, using the ideals defined from mod...

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is ...

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 P n−1. The main tool we use is a comparison between an easily obtained embedding into P

The “weak” BSD conjecture predicts that for an elliptic curve E over a number field K, we have rankE(K) = ords=1 L(E/K, s). The miracle of this formula is that it relates two quantities with very different origins: the left hand side is an algebraic object while the right hand side is an analytic object. Furthermore, the algebraic rank is “global” in nature, while the analytic rank can be defin...

In the past decade or so, the most elementary of the sieve methods of analytic number theory has been adapted to a geometric setting. In this geometric setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over Z. We will present the method and some of the results that have been proved using it. For instance, the prob...