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.     

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.     

Torsion points on families of products of elliptic curves

In a recent paper we proved 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. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.     

Torsion points on elliptic curves over fields of low degree

Partially supported by an N.S.A. grant     

Torsion points on modular curves

Let N≥23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the ℚ-valued points of the modular curve X ₀(N) which map to torsion points on J ₀(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X ₀(N) into J ₀(N). Oblatum 1-VI-1999 & 19-X-1999?Published online: 29 March 2000     

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 on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .

     

Torsion points of elliptic curves over large algebraic extensions of finitely generated fields

The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ ₁, …,σ _e)of elements of the abstract Galois group G(K)of K we have:

1. If e=1,then E _tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l.
2. If e≧2,then E _tor(K(σ))is finite.
3. If e≧1,then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order.

HereK(σ) is the fixed field in the algebraic closureK ofK, ofσ ₁, …,σ _e, and “almost all” is meant in the sense of the Haar measure ofG(K).     

Reductions of points on elliptic curves

Let E be an elliptic curve defined over \mathbbQ{\mathbb{Q}}. Let Γ be a subgroup of rank r of the group of rational points E(\mathbbQ){E(\mathbb{Q})} of E. For any prime p of good reduction, let [`(G)]{\bar{\Gamma}} be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have

Multiples of integral points on elliptic curves

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P∈E(Q), nP is integral for at most one value of n>C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E^′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ⊆E^′(Q), Γ contains at most 6 integral points. Explicit computations for congruent number curves are included.     

p-adicL-functions and rational points on elliptic curves with complex multiplication

Oblatum May 23, 1991     

Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

Height uniformity for integral points on elliptic curves

We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

     

Pair correlation of torsion points on elliptic curves

We prove the existence of the pair correlation measure associated to torsion points on the real locus E(R) of an elliptic curve E and provide an explicit formula for the limiting pair correlation function.     

On points of finite order on elliptic curves

A determination is made of all elliptic curves which possess, over a field K, points of order 10.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 563–567, May, 1970.