Torsion points on elliptic curves over fields of low degree

Partially supported by an N.S.A. grant     

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F   is a global function field of characteristic p>3$p>3$, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.     

Torsion points on elliptic curves over a global field

Let C be an elliptic curve defined over a global field K and denote by C_K the group of rational points of C over K. The classical Nagell-Lutz-Cassels theorem states, in the case of an algebraic number field K as groud field, a necessary condition for a point in C_K to be a torsion point, i.e. a point of finite order. We shall prove here two generalized and strongthened versions of this classical result, one in the case where K is an algebraic number field and another one in the case where K is an algebraic function field. The theorem in the number field case turns out to be particularly useful for actually computing torsion points on given families of elliptic curves.     

PAC fields over finitely generated fields

We prove the following theorem for a finitely generated field K: Let M be a Galois extension of K which is not separably closed. Then M is not PAC over K. Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation. This work constitutes a part of the Ph.D. dissertation of L. Bary-Soroker done at Tel Aviv University under the supervision of Prof. Dan Haran.     

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a -tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal.     

Primitive points on constant elliptic curves over function fields

Letf:CE be a non-constant rational map between curves over a finite field, whereE is elliptic. We estimate the number of rational points ofC whose image underf generate the group of rational points ofE.     

Ray class fields generated by torsion points of certain elliptic curves

We first normalize the derivative Weierstrass ???-function appearing in the Weierstrass equations which give rise to analytic parametrizations of elliptic curves, by the Dedekind ??-function. And, by making use of this normalization of ???, we associate a certain elliptic curve to a given imaginary quadratic field K and then generate an infinite family of ray class fields over K by adjoining to K torsion points of such an elliptic curve. We further construct some ray class invariants of imaginary quadratic fields by utilizing singular values of the normalization of ???, as the y-coordinate in the Weierstrass equation of this elliptic curve, which would be a partial result towards the Lang?CSchertz conjecture of constructing ray class fields over K by means of the Siegel?CRamachandra invariant.     

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P _λ of infinite order in the Mordell-Weil group of the elliptic curve Y ² = X ³ − n ² X over ℚ(λ). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain).     

Torsion points on elliptic curves with given j-invariant

Let E be an elliptic curve defined overQ, and let T(E) denote the group ofQ-rational torsion points on E. In this article an explicit method for computing T(E) for all E with a given j-invariant j is given. In particular, if j≠0, 2⁶ 3³ and E is defined by Y²=X³+AD²X+BD³ put into standard form with D its minimal D-factor, then a necessary condition that E possessQ-rational torsion points of order greater than 2 is that D|(2²A³+3³B²).     

Arithmetic height functions over finitely generated fields

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). Oblatum 7-VI-1999 & 21-IX-1999 / Published online: 24 January 2000     

Torsion points on families of squares of elliptic curves

In a recent paper we proved that there are at most finitely many complex numbers λ ≠  0,1 such that the points \({(2,\sqrt{2(2-\lambda)})}\) and \({(3, \sqrt{6(3-\lambda)})}\) are both torsion on the elliptic curve defined by Y ² = X(X ? 1)(X ? λ). Here we give a generalization to any two points with coordinates algebraic over the field Q(λ) and even over C(λ). This implies a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes.