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Tuesday, August 11, 2020 | History

3 edition of Elliptic curves over number fields with prescribed reduction type found in the catalog.

Elliptic curves over number fields with prescribed reduction type

Michael Laska

Elliptic curves over number fields with prescribed reduction type

by Michael Laska

  • 199 Want to read
  • 16 Currently reading

Published by Friedr. Vieweg & Sohn in Braunschweig .
Written in English

    Subjects:
  • Algebraic fields.,
  • Curves, Elliptic.

  • Edition Notes

    Includes bibliographical references and indexes.

    StatementMichael Laska.
    SeriesAspects of mathematics. E ;, vol. 4 =, Aspekte der Mathematik, Aspects of mathematics., v. 4.
    Classifications
    LC ClassificationsQA247 .L37 1983
    The Physical Object
    Paginationvi, 213 p. :
    Number of Pages213
    ID Numbers
    Open LibraryOL2900058M
    ISBN 10352808569X
    LC Control Number84128011

      The estimates then follow from elementary multiplicative number theory. In addition, we obtain infinite families of real and imaginary quadratic fields such that there are no elliptic curves with everywhere good reduction over these : Amanda Clemm, Sarah Trebat-Leder. Elliptic Curves over Finite Fields 1 B. Sury 1. Introduction Jacobi was the rst person to suggest (in ) using the group law on a cubic curve E. The chord-tangent method does give rise to a group law if a point is xed as the zero element. This can be done over any eld over which there is a rational point.

    Elliptic curves over a general field. Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written. ELLIPTIC CURVES OVER FINITE FIELDS. This chapter describes the specialised facilities for elliptic curves defined over finite fields. Details concerning their construction, arithmetic and basic properties may be found in Chapter ELLIPTIC of the machinery has been constructed with Elliptic Curve Cryptography in mind.

    Elliptic Curves Booksurge Publishing, pages, ISBN (ISBN is for the softcover version). This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. Softcover version available from bookstores worldwide. List price 17 USD; an online bookstore. At the primes of good reduction, the Euler factors coincide with those already constructed by Hasse-Weil, and at the primes of bad reduction it gives the Euler factors which you have written down. Most of this can be found in Silverman's book(s) on elliptic curves.


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Elliptic curves over number fields with prescribed reduction type by Michael Laska Download PDF EPUB FB2

Elliptic Curves over Number Fields with Prescribed Reduction Type. Authors: Laska, Michael Free PreviewPrice: $ Elliptic Curves over Number Fields with Prescribed Reduction Type. Authors; Michael Laska Search within book. Front Matter. Pages I-VI. PDF. Introduction. Michael Laska.

Pages Reduction of elliptic curves. Michael Laska. Pages Elliptic curves with good reduction outside a given set of prime ideals. Michael Laska. Pages The. A highly interesting topic that is included in the book concerns Neron models, which the author motivates by considering an elliptic curve E over the p-adic number field Q(p).

A change of variables to its Weierstrass equation is made so that ord() takes on its minimal value and the coefficients are in Z(p)/5(3). Elliptic curves with prescribed good reduction. Construction of elliptic curves with good reduction outside a finite set of primes.

A theorem of Shafarevich states that, over a number field \(K\), given any finite set \(S\) of primes of \(K\), there are (up to isomorphism) only a finite set of elliptic curves defined over \(K\) with good reduction at all primes outside \(S\). Elliptic curves over number fields. An elliptic curve \(E\) over a number field \(K\) can be given by a Weierstrass equation whose coefficients lie in \(K\) or by using base_extend on an elliptic curve defined over a subfield.

One major difference to elliptic curves over \(\QQ\) is that there might not exist a global minimal equation over \(K\), when \(K\) does not have class number one.

Reduction is a useful skill to have as an arithmetic geometer. Here we examine some elliptic curves whose reductions can be described relatively easily, and at the end some curious behavior of reduction mod upon extending the base field. Nearly all of this is from the book A First Course in Modular Forms by Diamond & Shurman, and in particular exercises and 1.

Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6.

Descent 22 7. Elliptic Units 27 8. Euler Systems 37 9. Bounding Ideal Class Groups 43 The Theorem of Coates and Wiles 47 Iwasawa Theory and. Let E over Q be an elliptic curve and let m be a positive integer and p a prime number such that gcd(p;m)=1.

For E modulo p the reduction map modulo p E(Q)[m]!E0(Z=p) is injective. Corollary The number of m-torsion points of E over Q divides the number of points over Z= Size: KB. An Introduction to the Theory of Elliptic Curves The Discrete Logarithm Problem Fix a group G and an element g 2 Discrete Logarithm Problem (DLP) for G is: Given an element h in the subgroup generated by g, flnd an integer m satisfying h = gm: The smallest integer m satisfying h = gm is called the logarithm (or index) of h with respect to g, and is denotedFile Size: KB.

Part II. The Algebraic Theory of Elliptic Curves 4. The field of rational functions on a curve The key to dealing with algebraic curves over a ground field that is not alge-braically closed is to abandon the notion of points of a curve and to work instead with rational functions on the curve.

These rational functions form a field, the. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers.

Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or Cited by: Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field.

It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality by: $\begingroup$ The formation of the Neron model over henselian discrete valuation rings commutes with scalar extension to the maximal unramified extension and its completion, so for any "table" of reduction types it is often sufficient to consider only separably closed residue fields.

Hence, to the extent the residue field is perfect, it usually may as well be algebraically closed. Abramovich. Formal finiteness and the torsion conjecture on elliptic curves. A footnote to a paper: “Rational torsion of prime order in elliptic curves over number fields” [Astérisque No.

(), 3, 81–] by S. Kamienny and B. Mazur. Astérisque, ():3, 5–17, Columbia University Number Theory Seminar (New York, ).Author: Joseph H. Silverman. Vol NumberPages { S (99) Article electronically published on REDUCTION OF ELLIPTIC CURVES OVER CERTAIN REAL QUADRATIC NUMBER FIELDS MASANARI KIDA Abstract.

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic eld has a 2-rational point under.

Additional Physical Format: Online version: Laska, Michael. Elliptic curves over number fields with prescribed reduction type. Braunschweig: Friedr. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call \emph{false complex multiplication}, have even by: There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields.

We present examples of high rank elliptic curves with given torsion group which give the. MATHEMATICS OF COMPUTATION, VOL NUMBER JULYPAGES Elliptic Curves Over Finite Fields.

II By I. Borosh, C. Moreno and H. Porta Abstract. The class groups of certain elliptic function fields without complex multiplica-tions are computed. Questions about the structure of these groups and the arithmetical. Keywords Torsion Group, Elliptic Curves, Number Fields Mathematics Subject Classi cation () 11G05, 11G18, 14H52 1 Introduction Our goal is to study the number of elliptic curves over number elds of prime degree K=Q, up to K-isomorphism, having a prescribed property.

This property will be one of the following: 1) The curve’s group of K. ranks of elliptic curves with prescribed torsion over number fields 7 and those of the form Z / 2 Z ⊕ T ′, where T ′ occur s over a quadratic field and has exactly one element of order 2.We construct infinite families of elliptic curves with cyclic torsion groups over quartic number fields K such that the Galois closure of K is dihedral of degree 8; such a quartic number field K is called a dihedral quartic number fact, all the cyclic torsion groups of elliptic curves which occur over quartic number fields (but not over quadratic number fields) are Z / N Z with N = 17 Cited by: 4.Additive reduction of elliptic curves.

Ask Question Asked 10 years ago. Active 10 years ago. Ramification in p-division fields associated to elliptic curves with good ordinary reduction. Surjectivity of reduction maps of elliptic curves over Q.

4.