Group Of Rational Points Research Articles

Classical Invariants and 2-descent on Elliptic Curves

The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented in Birch and Swinnerton-Dyer (1963), and can be applied to give a more efficient algorithm for determining Mordell–Weil groups over Q, as well as being more readily extended to other number fields. In this paper we mainly restrict ourselves to general theory, valid over arbitrary fields of characteristic neither 2 nor 3.

On the Cyclicity of Elliptic Curves over Finite Field Extensions

It is well known that elliptic curves with cyclic group of rational points over nite elds play an important role in many applications of such curves to artithmetic, cryptography, etc. (cf., for example, [1, 2, 6]). It is but natural to ask for elliptic curve (with a cyclic group of rational points over a nite eld) to to describe the extensions of the ground eld for which the curve is cyclic (i.e. has a cyclic rational points group) as well. This question is Problem 6.3 in [6]. Let us x some notation and pose the problem more precisely. Let E be an elliptic curve over a nite eld F q with qpa elements, its characteristic being p; let E(F q ) be the group of its F q -rational points. We are interested in the set

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

There are well-known subexponential algorithms for finding discrete logarithms over finite fields. However, the methods which underlie these algorithms do not appear to be easily adaptable for finding discrete logarithms in the groups associated with elliptic curves and the Jacobians of hyperelliptic curves, except for very special cases (Menezes et al., IEEE Trans. Inform. Theory 39 (1993) 1639–1646, Okamoto and Sakurai, Lecture Notes in Computer Science, vol. 576, Springer, Berlin, 1991). This has led to the development of cryptographic systems based on the discrete logarithm problem for such groups (Koblitz, Math. Comput. 48 (1987) 203–209, J. Cryptogr. 1 (1989) 139–150, Menezes, Elliptic Curve Public Key Cryptosystems, Kluwer, Dordrecht, 1993, Miller, Lecture Notes in Computer Science, Springer, Berlin, 1986, pp. 417– 426). In this paper a subexponential algorithm is presented for finding discrete logarithms in the group of rational points on the Jacobians of large genus hyperelliptic curves over prime fields. We give a heuristic argument that under certain assumptions, there exists a constant c⩽2.181 such that for g∈ Z >0 and odd prime powers q with log q⩽(2 g+1) 1− ε and 0< ε=o(1), the algorithm computes discrete logarithms in the group of rational points on the Jacobian of a genus g hyperelliptic curve over GF(q) of the form y 2= f( x) with deg( f)=2 g+1, within expected time L q 2 g+1 [1/2, c], or equivalently, L q g [1/2, c′] with c′= 2 c .

Representations of abelian algebraic groups

There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension

The Group of Rational Points on the Unit Circle

There have been several attempts to present the basic idea of this marvelous proof to a wider mathematical audience ([2, 7, 81). Another recent paper [31 provides more details. Wiles' proof is based on the theory of elliptic curves, i.e., curves defined by cubic equations. A big part of this theory is devoted to understanding the (points whose coordinates are rational numbers) on these curves. The set of rational points on an elliptic curve has a natural group structure, which will be described very briefly later. It is often very difficult, however, to find all the rational points on an elliptic curve. In this paper we take a much easier and more familiar example-the unit circle-and show how to compute the group structure of its rational points. Next, some applications are given. Finally, for comparison, we give a brief summary of known results on the 'group structure for rational points on elliptic curves.

On a Class of Elliptic Curves with Rank at Most Two

In this note we consider the elliptic curves y 2 = x 3 + px defined over Q for primes p satisfying p ? 1 (mod 8), and review some of their properties. We then compute and list (in the supplement) their ranks, and give, when the rank is positive, the generators of the group of rational points and Mordell-Weil lattice invariant τ for all primes p < 50000 of the form m 2 + 64n 2 .

On rational-derived quartics

We present a characterisation of all quartic polynomials with exactly three distinct roots and the property that it and all its derivatives have rational roots. It turns out that there are an infinite number of distinct such quartics, each of which corresponds to a point on a related elliptic curve. Furthermore the collection of these points forms a proper subgroup of the group of rational points on the curve.

On ranks of twists of elliptic curves and power-free values of binary forms

Let E be an elliptic curve over Q. By the rank of E we shall mean the rank of the group of rational points of E. Mestre [31], improving on the work of Neron [34] (cf. [13], [39] and [46]), has shown that there is an infinite family of elliptic curves over Q with rank at least 12. However, computational work (see [3], [4], [7] and [47]) suggests that a typical elliptic curve will have much smaller rank, with curves of rank 0 or 1 being predominant. Indeed, Brumer [6] has proved, subject to the Birch, Swinnerton-Dyer conjecture, the Shimura, Taniyama, Weil conjecture and the generalized Riemann hypothesis, that the average rank of an elliptic curve, ordered according to its Faltings height, is at most 2.3. In this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over Q. That is, we shall restrict our attention to families of elliptic curves defined over Q which are isomorphic over C. There are families of quadratic, cubic, quartic and sextic twists (see, for example, Proposition 5.4 of Chapter X of [42]). Let E be an elliptic curve over Q with Weierstrass equation y2 = X + ax + b and for any non-zero integer d let Ed 2 3 denote a quadratic twist of E given by the equation dy2 = X + ax + b. Let r(d) denote the rank of Ed . Note that if d1 and d2 are non-zero integers, then Ed is isomorphic to Ed over Q if and only if d1 /d2 is the square of a rational number. Subject to the conjectures of Birch and Swinnerton-Dyer and of Shimura, Taniyama, and Weil, Goldfeld [14] conjectured in 1979 that

On a class of elliptic curves with rank at most two

In this note we consider the elliptic curves y 2 = x 3 + p x {y^2} = {x^3} + px defined over Q \mathbb {Q} for primes p satisfying p ≡ 1 ( mod 8 ) p \equiv 1\; \pmod 8 , and review some of their properties. We then compute and list (in the supplement) their ranks, and give, when the rank is positive, the generators of the group of rational points and Mordell-Weil lattice invariant τ \tau for all primes p &gt; 50000 p &gt; 50000 of the form m 2 + 64 n 2 {m^2} + 64{n^2} .

On the Equation $Y^2=(X+p)(X^2+p^2)$

textabstractIn this paper the family of elliptic curves over Q given by the equation y2 = (x + p)(x2 + p2) is studied. It is shown that for p a prime number = ±3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2, Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p = 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p = 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve arc constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication. Copyright

Homomorphisms From the Group of Rational Points On Elliptic Curves to Class Groups of Quadratic Number Fields

The main purpose of this paper is to prove that there is a homomorphism from the group of primitive points on an elliptic curve given by an equation Y2 = X3 + a2X2 + a4X + a6 to the ideal class group of the order Z + Z [formula]. Two applications are given. First we prove a conjecture concerning the order of ideals coming from rational points of infinite order on the curve. Then we describe how to construct families of quadratic number fields containing a subgroup of the ideal class group isomorphic to the torsion group of the curve.

On some new local-global principles over a real function field

We prove here the validity of the cohomological Hasse principle for almost simple simply connected groups of classical type defined over the rational real function field R(t) and the Hasse principle (analog of the Platonov conjecture) about the projective simplicity of the groups of rational points of such groups.

Minimum distance of elliptic codes

In this paper we investigate three classes of linear codes arising from elliptic curves and compute their minimal distance using implicity the structure of the group of rational points of the curves.

The characters of the group of rational points of a reductive group with non-connected centre.

The characters of the group of rational points of a reductive group with non-connected centre.

The square-free sieve and the rank of elliptic curves

Let E be an elliptic curve over Q. A celebrated theorem of Mordell asserts that E(Q), the (abelian) group of rational points of E, is finitely generated. By the rank of E we mean the rank of E(Q). Thus the rank of E is positive if and only if E possesses an infinity of rational points. Relatively few general qualitative assertions can be made about the rank as E varies. How large can the rank get? Although we expect that there are elliptic curves over Q with arbitrarily high ranks, this is presently unknown. What is the average size of the rank? Recently Brumer and McGuinness have reported on their study of 310,716 elliptic curves of prime conductor less than 108 where they have found that 20.06% of those curves have even rank > 2. (Cf. [BM]; also see forthcoming work of Brumer where, subject to a number of standard conjectures, he shows that 2.3 is an upper bound for the average rank for all elliptic curves over Q ordered in terms of their Faltings height.) What is the behavior of the rank over the family of twists of a given elliptic curve? Here, one has three natural kinds of families of twists: (1) Quadratic twists. One can take any elliptic curve and systematically twist it by all quadratic characters (this is the type of family of elliptic curves we are concerned with in this paper). Specifically, if E is an elliptic curve over Q given by the Weierstrass equation Y2 = X + A * X + B, and D is any squarefree integer, the (quadratic) twist of E by D, ED, is given by the equation

Structure of groups of rational points of classical algebraic groups over number fields

Structure of groups of rational points of classical algebraic groups over number fields

Projective simplicity of groups of rational points of simply connected algebraic groups defined over number fields

Projective simplicity of groups of rational points of simply connected algebraic groups defined over number fields

Arithmetic distance functions and height functions in diophantine geometry

In virtually all areas of Diophantine Geometry, the theory of height functions plays a crucial role. This theory associates to each (Cartier) divisor D on a projective variety V a function ho mapping the group of rational points of V to the real numbers. This function, which is defined up to a bounded function on V, has very nice functorial properties. For example, it is linear in D and depends only on the linear equivalence class of D. As a consequence, any relation between divisor classes, such as the theorem of the square for abelian varieties, will yield a corresponding relation for height functions. Further, it is possible to write the height function ho as a sum of local height functions 20(" ; v), where v ranges over the distinct absolute values of the given field. (These local heights are also known as logarithmic distance functions or Weil functions.) Each 2 o is defined away from the support of D, and gives a measure of the v-adic distance from the given point of V to the divisor D. In this paper we propose to define local height functions 2 x for every closed subscheme X of a projective variety V. As above, 2x(; v) will give a measure of the v-adic distance from a point of V to X. These functions will have many nice functorial properties; for example, 2xny will equal the minimum of 2x and hr. In this way, relations between closed subschemes (or equivalently, between ideal sheaves) will yield relations between local height functions. We will thus be able to convert geometric statements into arithmetic statements relatively painlessly. More generally, we will deal with the case that V is merely quasi-projective by defining a function 20v('; v) which will measure the v-adic distance to the "boundary" of V. We will then be able to assign to each closed subscheme X of V a local height function 2x which will be well defined up to addition of a multiple of 2~v. This extra generality is useful, for example, when one has a complete family of varieties and wishes to discard the "bad" fibers. As one application of the machinery that we have developed, we prove a quantitative version of the inverse function theorem. Thus given a finite map, we use local height functions to describe how far away from the ramification locus

On the equation 𝑌²=𝑋(𝑋²+𝑝)

Generators are found for the group of rational points on the title curve for all primes p ≡ 5 ( mod 8 ) p \equiv 5\;\pmod 8 less than 1,000. The rank is always 1 in accordance with conjectures of Selmer and Mordell. Some of the generators are rather large.

Generators, relations and coverings of algebraic groups, II

THEOREM 1.1. Let G be the group of rational points of a simply connected simple afine algebraic group defined and split over a fmite j?eld of q elements. Then the Schur multiplier M of G is trivial with the following exceptions: (a) Zf G is A,(4), A,(2), A,(3), C,(2), F,(2) or G,(4) then h4 is Z, (cyclic of order 2); if G is A,(9), B,(3) or G,(3) it is iz,. (b) Zf G is A,(4) (resp. D,(2)) then M is the direct product of Z, (resp. Z,) with itself.