Horst G. Zimmer

Constructing elliptic curves with given group order over large finite fields

A procedure is developed for constructing elliptic curves with given group order over large finite fields. The generality of the construction allows an arbitrary choice of the parameters involved. For instance, it is possible to specify the finite field, the group order or the class number of the endomorphism ring of the elliptic curve. This is important for various applications in computational number theory and cryptography. Moreover, we give a method that yields all representations of a given integer as a norm in an imaginary quadratic field.

On Mordell's equation

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordells Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordells Equation in Z for all integers k within the range 0 < | k | ≤ 10 000 and partially extend the computations to 0 < | k | ≤ 100 000. For these values of k, the constant in Halls conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

Computing all S -integral points on elliptic curves

In this note we combine the advantages of the methods of Siegel-Baker-Coates and of Lang-Zagier for the computation of S-integral points on elliptic curves in Weierstrass normal form over the rationals. In this way we are able to overcome the absence of an explicit lower bound for linear forms in q-adic elliptic logarithms. We present an efficient algorithm for determining all S-integral points on such curves.

Elliptic curves : a computational approach

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.

Torsion groups of elliptic curves with integral j-invariant over pure cubic fields

Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Communicated by P. Roquette Received December 28, 1988; revised October 3, 1989 We determine all possible torsion groups of elliptic curves E with integral j-invariant over pure cubic number fields K. Except for the groups Z/22, Z/32 and Z/22 @ Z/22, there exist only finitely many curves E and pure cubic fields K such that E over K has a given torsion group E,, (K), and they are all calculated here. The curves E over K with torsion group E roOR( K) r Z/22 0 Z/22 have j-invariants belonging to a finite set. They are also calculated. A preliminary report on the results obtained was given by H. H. Miiller, H. Stroher, and H. G. Zimmer (in “Proceedings, Intern. Numb. Th. Conference at Lava1 University, Quebec, Canada, 1987” (J.-M. DeKoninck and C. Levesque, Eds.), pp. 671-698, de Gruyter, Berlin/ New York, 1989).

Torsion Groups of Elliptic Curves with Integral j-Invariant over General Cubic Number Fields

In [15] and [16] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion...

An algorithm for determining the regulator and the fundamental unit of hyperelliptic congruence function field

Let K be an algebraic function field in one indeterminate X over a finite field of constants k = IFq of characteristic p. Hence K/k is a congruence function field of transcendence degree one. Suppose that K/k is elliptic or hyperelliptic and that k has characteristic p ~ 2. Then K is a quadratic extension of the rational subfield K. . k(X). In fact there IS a square-free polynomial D c k[X] such that K = KO(~D ), and the integral closure of the polynomial ring k[X~ in K is R = k[X ,<D ]. Suppose moreover, that K/KO is a real quadratic extension, i.e. that the place at infinity pm of K. with respect to X splits in K: pa = plpz. ln analogy to the case of a real quadratic number field, the unit group U of the real quadratic function field K/k is the direct product of the finite cyclic group kx of non-zero constants and an infinite cyclic group <E> generated by the fundamental unit E of K: U = k’ x <E> . The absolute value of the normalized additive valuation Vp of E associated with pi, i.e. r =

On fermat's quadruple equations

A. DUJELLA [2] proved that, if n ~ 2 (mod 4) and n r { -4 , 3 , 1 , 3, 5, 8, 12, 20}, then there exists at least one D(n)-quadruple consisting of positive integers only. Moreover, DUJELLA showed in [3] that every D(1)-quadruple of integers can be extended to a D(1)-quintuple of rational numbers. More generally, if we are given a set of rational numbers {bl . . . . . bl} _c Q satisfying the system of equations

Computing S-Integral Points on Elliptic Curves

By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demjanenko [L3] states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4]). This conjecture was proved by Silverman [Si1] for elliptic curves E with integral modular invariant j over K and by Hindry and Silverman [HSi] for algebraic function fields K. On the other hand, beginning with Baker [B], bounds for the size of the coefficients of integral points on E have been found by various authors (see [L4]). The most recent bound was exhibited by W. Schmidt [Sch, Th. 2]. However, the bounds are rather large and therefore can be used only for soloving some particular equations (see [TdW], [St]) or for treating a special model of elliptic curves, namely Thue curves of degree 3 (see [GSch]).

Computational number theory : proceedings of the Colloquium on Computational Number Theory held at Kossuth Lajos University, Debrecen (Hungary), September 4-9, 1989

The volume is devoted to the interaction of modern scientific computation and classical number theory. The contributions, ranging from effective finiteness results to efficient algorithms in elementary, analytical and algebraic number theory, provide a broad view of the methods and results encountered in the new and rapidly developing area of computational number theory. Topics covered include finite fields, quadratic forms, number fields, modular forms, elliptic curves and diophantine equations. In addition, two new number theoretical software packages, KANT and SIMATH, are described in detail with emphasis on algorithms in algebraic number theory.