No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle
... As we will need to employ known results on cubic extensions of quadratic number fields, we recall below some known facts and definitions for completeness. The interested reader may refer to [6,Chapter 5] and [10,Chapter 4] for more details, and to [1] for a short account. ...
We consider all \emph{odd} fundamental discriminants and their mirror discriminants , and we study the family of elliptic curves . We denote by and the rank of the 3-part of the ideal class group of and respectively. We show that every curve in the subfamily of elliptic curves with for (respectively, with for ) cannot have any integral points, and this is proved unconditionally. By employing results of Satg\'e and by assuming finiteness of the 3-primary part of their Tate-Shafarevich group, we show that the curves must have odd rank when and even rank when . This result is particularly interesting for the case of since every curve with has infinitely many rational points - assuming finiteness of the 3-primary part of their Tate-Shafarevich group - yet no integral points. We obtain an unconditional result on the existence of elliptic curves with non-trivial rank and no integral points, by defining a parametrised family of such curves with no integral points but with a parametrised rational point, which we prove that it is of infinite order.
We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.
Recall that an order O in an algebraic number field K is called monogenic if it is generated by one element, i.e., there is an α with ℤ[α] = O. By work of Gyo″ry [11, 1976] there are, up to a suitable equivalence, only finitely many α such that ℤ[α] = O. In this survey, we give an overview of recent results on estimates for the number of α up to equivalence.
The discriminant of a trinomial of the form has the form
if n and m are relatively prime. We
investigate when these discriminants have nontrivial square factors. We explain
various unlikely-seeming parametric families of square factors of these
discriminant values: for example, when n is congruent to 2 (mod 6) we have
that always divides . In addition, we
discover many other square factors of these discriminants that do not fit into
these parametric families. The set of primes whose squares can divide these
sporadic values as n varies seems to be independent of m, and this set can
be seen as a generalization of the Wieferich primes, those primes p such that
is congruent to 1 (mod ). We provide heuristics for the density
of squarefree values of these discriminants and the density of these "sporadic"
primes.
We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 10 11 and complex cubic fields down to −10 11 have been computed. The classification of quadratic fields up to isomorphism is trivial: they are uniquely characterized by their discriminant, and we can compute tables as soon as we know how to test if an integer is squarefree and how to check some simple congruence modulo 16. We intend to show that cubic fields are essentially as easy to deal with, and we will get a canonical representation for them. Contrary to the quadratic case, the treatment depends on the signature but, the fundamental ideas being the same, we shall expose as much as we can before splitting cases. Almost all results in this paper are either ancient or elementary. I would like to thank Professor H. Cohen for his interest when I first mentioned what I thought was a trivial application of some well known results. Moreover, his careful reading of successive drafts of this work and the many questions he had about it were most helpful in giving it its present shape.
This is the third in a series of papers in which we study the n-Selmer group
of an elliptic curve, with the aim of representing its elements as curves of
degree n in P^{n-1}. The methods we describe are practical in the case n=3 for
elliptic curves over the rationals, and have been implemented in Magma.
One important ingredient of our work is an algorithm for trivialising central
simple algebras. This is of independent interest: for example, it could be used
for parametrising Brauer-Severi surfaces.
We give a three-descent procedure to bound and, in some cases, compute the three-part of the Selmer and Tate-Shafarevich group of the curves y2 =x3 + a, a a nonzero integer. This enables us to verify the whole Birch and Swinnerton-Dyer conjecture for some of such curves.
A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over Q. Remarks are given concerning the extension of these results to forms defined over number fields. This paper has now appeared in the LMS Journal of Computation and Mathematics, Volume 2, pages 62--92, and the full published version, with hyperlinks etc., is available online (to subscribers) at http://www.lms.ac.uk/jcm/2/lms98007/. This version contains the complete text of the published version in a very similar format. 1. Introduction Reduction theory for polynomials has a long history and numerous applications, some of which have grown considerably in importance in recent years with the growth of algorithmic an...
This is the first in a series of papers in which we study the n-Selmer group
of an elliptic curve, with the aim of representing its elements as genus one
normal curves of degree n. The methods we describe are practical in the case
n=3 for elliptic curves over the rationals, and have been implemented in Magma.
The key feature at this conference was the 33 invited papers from the world's leading number theorists. Talks were in three sessions: analytical number theory; arithmetical algebraic geometry; and diophantive approximation. Speakers included: F.Beukers (University of Utrecht); R. Heath-Brown (Oxford); H.L. Montgomery (Ann Arbor, Michigan); T. Nakahara (Saga University, Japan); Y. Zarhin (Academy of Science, USSR).
In this paper, we describe an algorithm that reduces the computation of the (full) p p -Selmer group of an elliptic curve E E over a number field to standard number field computations such as determining the ( p p -torsion of) the S S -class group and a basis of the S S -units modulo p p th powers for a suitable set S S of primes. In particular, we give a result reducing this set S S of ‘bad primes’ to a very small set, which in many cases only contains the primes above p p . As of today, this provides a feasible algorithm for performing a full 3 3 -descent on an elliptic curve over Q \mathbb Q , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of E [ p ] E[p] is favorable, simplifications are possible and p p -descents for larger p p are accessible even today. To demonstrate how the method works, several worked examples are included.
Let K be a number field with ring of integers O K . K is said to be monogenic if O K = Z[θ] for some θ ∈ O K . Monogeneity of a number field is not always guaranteed. Furthermore, it is rare for two number fields to have the same discriminant, thus finding fields with these two properties is an interesting problem. In this paper we show that there exist infinitely many triples of polynomials defining distinct monogenic cubic fields with the same discriminant. © Société Arithmétique de Bordeaux, 2018, tous droits réservés.
For any nonzero h ∈ ℤ, we prove that a positive proportion of integral binary cubic forms F do locally everywhere represent h but do not globally represent h; that is, a positive proportion of cubic Thue equations F(x,y) = h fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms F by their absolute discriminants. We prove the same result for Thue equations G(x,y) = h of any fixed degree n ≥ 3, provided that these integral binary n-ic forms G are ordered by the maximum of the absolute values of their coefficients.
The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry.
Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of this book. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra’s elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat’s Last Theorem by Wiles et al. via the use of elliptic curves.
The equations show the first prime numbers that can be represented as a sum of two squares. Except for 2, they are all ≡ 1 mod 4, and it is true in general that any odd prime number of the form p = a 2 + b 2 satisfies p ≡ 1 mod 4, because perfect squares are ≡ 0 or ≡ 1 mod 4. This is obvious.
In this article we shall give a survey of Hasse’s problem for integral power bases of algebraic number fields during the last half of century. Specifically, we developed this problem for the abelian number fields and we shall show several substantial examples for our main theorem [7] [9], which will indicate the actual method to generalize for the forthcoming theme on Hasse’s problem [15].
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points.
For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.
The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.
In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X0(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p⩽7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.
Soit k un entier rationnel, et soit A la courbe elliptique d'équation projective Y2Z = X3 + kZ3. Le quotient de A par son sous-groupe engendré par le point (0, , 1) est la courbe A′ d'équation projective Y2Z = X3 − 27kZ3. Nous calculons le groupe de Selmer sur de la projection de A sur A′ qui est une isogénie de degré 3. Nous utilisons ce calcul pour retrouver et améliorer des résultats classiques sur les courbes du type Y2Z = X3 + kZ3, courbes qui ont été l'objet de nombreux travaux.
We give a new proof of Voronoi's determination of an integral basis for a cubic field. Let K be a cubic field. Without loss of generality we may take the cubic field K in the form K = Q(µ), where µ is a root of the irreducible polynomial f(x) = x3 ¡ ax + b , a,b 2 Z. For each prime p and each nonzero integer m, "p(m) denotes the greatest exponent l such that pl | m. We can also assume that for every prime p "p(a) < 2 or "p(b) < 3 , p>3 1•"p(b)•"p(a) p2, where fi and fl are given by
An asymptotic formula is proved for the number of cubic fields of discriminant delta in 0 < delta < X; and in -X < delta < 0.
In the rst of two papers on Magma, a new system for computational algebra, we present the Magma language, outline the design principles and theoretical background, and indicate its scope and use. Particular attention is given to the constructors for structures, maps, and sets. c 1997 Academic Press Limited Magma is a new software system for computational algebra, the design of which is based on the twin concepts of algebraic structure and morphism. The design is intended to provide a mathematically rigorous environment for computing with algebraic struc- tures (groups, rings, elds, modules and algebras), geometric structures (varieties, special curves) and combinatorial structures (graphs, designs and codes). The philosophy underlying the design of Magma is based on concepts from Universal Algebra and Category Theory. Key ideas from these two areas provide the basis for a gen- eral scheme for the specication and representation of mathematical structures. The user language includes three important groups of constructors that realize the philosophy in syntactic terms: structure constructors, map constructors and set constructors. The util- ity of Magma as a mathematical tool derives from the combination of its language with an extensive kernel of highly ecient C implementations of the fundamental algorithms for most branches of computational algebra. In this paper we outline the philosophy of the Magma design and show how it may be used to develop an algebraic programming paradigm for language design. In a second paper we will show how our design philoso- phy allows us to realize natural computational \environments" for dierent branches of algebra. An early discussion of the design of Magma may be found in Butler and Cannon (1989, 1990). A terse overview of the language together with a discussion of some of the implementation issues may be found in Bosma et al. (1994).
For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we
give an unconditional construction of infinitely many monic irreducible
polynomials of degree n with integer coefficients having squarefree
discriminant and exactly r real roots. These give rise to number fields of
degree n, signature (r,s), Galois group S_n, and squarefree discriminant; we
may also force the discriminant to be coprime to any given integer. The number
of fields produced with discriminant in the range [-N, N] is at least c
N^(1/(n-1)). A corollary is that for each n \geq 3, infinitely many quadratic
number fields admit everywhere unramified degree n extensions whose normal
closures have Galois group A_n. This generalizes results of Yamamura, who
treats the case n = 5, and Uchida and Yamamoto, who allow general n but do not
control the real place.
We give simple proofs of the Davenport--Heilbronn theorems, which provide the
main terms in the asymptotics for the number of cubic fields having bounded
discriminant and for the number of 3-torsion elements in the class groups of
quadratic fields having bounded discriminant. We also establish second main
terms for these theorems, thus proving a conjecture of Roberts. Our arguments
provide natural interpretations for the various constants appearing in these
theorems in terms of local masses of cubic rings.
Conference held Sept 1-17, 1965, at the University of Sussex, Brighton, Eng. Organized by the London Mathematical Society (a Nato advanced study institute) with the support of the International Mathematical Union Incluye referencias bibliográficas
User's Guide to PARI-GP (version 2.12.0-ALPHA)
- C Batut
- K Belabas
- D Bernardi
- H Cohen
- M Olivier
On a problem of Hasse, Algebraic number theory and related topics
- Y Motoda
- T Nakahara
- Ali Shah
- T Uehara