Search

Let C be an elliptic curve (an abelian variety of dimension one) defined over the field Q of rational numbers. A minimal Weierstrass model for C at all primes p in the sense of Néron [3] is given by a plane cubic equation of the formwhere aj belongs to the ring Z of integers of Q, the zero of C being the point of infinity.

Let E be an elliptic curve defined overQ. The group ofQ- rational points of finite order on E is a finite group T(E). In this article T(E) is computed for all elliptic curves defined overQ admitting complex multiplication. The only possible values for the order t of T(E) are 1, 2, 3, 4, or 6 in these cases. A standard form for an affine equation describing an elliptic curve with a given j-invariant is obtained. This is used to show that if j ≠ 0, 26 33, then the number ofQ- rational points of order 2 on E depends only on j. The results are summarized in the accompanying table.

It has long been known that every positive semidefinite function of R(x, y) is the sum of four squares. This paper gives the first example of such a function which is not expressible as the sum of three squares. The proof depends on the determination of the points on a certain elliptic curve defined over C(x). The 2-component of the Tate-Safarevic group of this curve is nontrivial and infinitely divisible.

n/a

It will be proved that under given conditions the torsion of the points of an elliptic curve over a number field is bounded.

Equality of distributions is shown of even and odd values of the order of the zero at the point s=1 of L-functions of elliptic curves x3+y3=D, where D is a positive integer not divisible by a cube.

In this paper we describe the intersection of the kernels of Manin homomorphisms for an arbitrary elliptic curve over a field of formal power series.

Starting from the well-known equation of Mandelstam’s boundary curve of a square diagram, we investigate the elliptic integral associated with this curve. Legendre’s modulusk2 depends exclusively on the ratiom/Μ of the two masses-the mass of the « nucleon » and of the « pion ». ForΜ=m andΜ = 2m, the genus of the curve is zero. A birational correspondence can be established in this case between the coordinates of the two curves so that the corresponding diagrams are, in a certain sense, equivalent. Finally, the values of the spectral density and scattering amplitude on the boundary curve are calculated. The scattering amplitude is given by an elliptic integral of the third kind.

n/a

We prove the following.Theorem. Let be a number field, and the Jacobian of the curve parametrizing the elliptic curves with distinguished cyclic subgroups of order . If the number is written as , where contains a -simple abelian subvariety such that {\operatorname{rk}} A_k,$ SRC=http://ej.iop.org/images/0025-5734/11/2/A12/tex_sm_2058_img8.gif/> then the set of -isomorphism classes of elliptic curves over the field possessing -points of order is finite.Bibliography: 4 items.

An explicit bound is obtained for the torsion of elliptic curves over the field of rational numbers. Let Γ be an elliptic curve over the field of rational numbers R, and Qm a primitive R-point of order m on it; here m is a prime or a double prime. Hence if m=2p, then p ≤ 509, whereas if m=p, then p < 6144.

A determination is made of all elliptic curves which possess, over a field K, points of order 10.

n/a

IF, in an elastic system with one degree of freedom, and friction proportional to the velocity, the relation of the “free” force to the displacement be considered, an interesting curve results.

In the Plücker formula for a curve embedded in a higher dimensional projective space, one encounters the notion of stationary point (cf, [B], [W]). W. F. Pohl gave new view point about it in terms of vector bundles and he defined “the singularities of embedding” (cf. [P]). At first, we shall give dual formulation of Pohl’s one by means of the sheaf of principal parts of order n, and next we shall prove the following: If an elliptic curve is embedded in (n — l)-dimensional projective space as a curve of degree n, singularities of projective embedding of order n — 1 are exactly the points of order n with suitable choice of a neutral element on the curve which is an abelian variety of dimension one. The proof is given by making use of the relation between and Schwarzenberger’s secant bundle which we shall also give.