Suppose p is a prime of the form u + 64 for some integer u, which we take to be 3 mod 4. Then there are two Neumann–Setzer elliptic curves E0 and E1 of prime conductor p, and both have Mordell–Weil group Z/2Z. There is a surjective map X0(p) π −→ E0 that does not factor through any other elliptic curve (i.e., π is optimal), where X0(p) is the modular curve of level p. Our main result is that th...

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...

We review previous methods of computing the modular degree of an elliptic curve, and present a new method (conditional in some cases), which is based upon the computation of a special value of the symmetric square L-function of the elliptic curve. Our method is sufficiently fast to allow large-scale experiments to be done. The data thus obtained on the arithmetic character of the modular degree...

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields Q( √ D), where D is a squarefree integer. For this purpose, we give a characterization in terms of Q( √ D)-rational points on the elliptic curve E : y = x − 27. We compute the torsion subgroup of the Mordell–Weil group of this elliptic curve over Q( √ D) and we give an explicit answer, i...

Let k be a global field, k a separable closure of k, and Gk the absolute Galois group Gal(k/k) of k over k. For every σ ∈ Gk, let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ Gk, the Mordell-Weil group E(k σ ) has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametriz...

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module an abelian variety. [G. Banaszak and P. Krasoń, On étale K-groups curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. author obtained sufficient condition validity [Formula: see text]-theory curve. This fact has been established by means analysi...

Given an elliptic curve E1 over a number field K and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2Selmer groups of E1, E2 and A, we can obtain information about X(E1/K)[2] and we give examples where...

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ odd prime number. $K_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $K_n$ denote its $n$-th layer. The Mordell--Weil is said to constant in tower if for all $n$, $E(K_n)$ equal $E(K)$. We apply techniques Iwasawa theory obtain explicit conditions above sense. then indicate potential applications Hilbert's ten...

In [Ro76], M. Rosen showed that for any countable commutative group G, there is a field K, an elliptic curve E/K and a surjective group homomorphism E(K) → G. From this he deduced that any countable commutative group whatsoever is the ideal class group of an elliptic Dedekind domain – the ring of all functions on an elliptic curve which are regular away from some (fixed, possibly infinite) set ...