The arithmetic of elliptic curves

@inproceedings{Silverman1986TheAO,
  title={The arithmetic of elliptic curves},
  author={Joseph H. Silverman},
  booktitle={Graduate texts in mathematics},
  year={1986}
}
  • J. Silverman
  • Published in Graduate texts in mathematics 1986
  • Mathematics, Computer Science
Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1). 
THE GEOMETRY OF ELLIPTIC CURVES OVER FINITE FIELDS
We first provide an overview of the basic results in the geometry of elliptic curves, introducing the Picard Group, Weierstrass Equations, and Isogenies. This is followed by a discussion of the
Topic In Elliptic Curves Over Finite Fields: The Groups of Points
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves
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The distribution of the quadratic residues at the x-coordinates of the sequence of points corresponding to progressions if the elliptic curves is defined over a simple field is established.
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This document presents some of the underlying theory and then summarize recent results concerning the aforementioned relationship between elliptic curves and graphs, and elucidated by theory that was omitted in their original presentation.
On the number of distinct elliptic curves in some families
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These formulas give explicit formulas for the number of distinct elliptic curves over a finite field in several families of curves of cryptographic interest such as Edwards curves and their generalization due to D. J. Bernstein and T. Lange.
Local root numbers of elliptic curves over dyadic fields
We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the
The Mordell-weil Theorem for Q
An elliptic curve is a specific type of algebraic curve on which one may impose the structure of an abelian group with many desirable properties. The study of elliptic curves has far-reaching
Explicit second p-descent on elliptic curves
One of the fundamental motivating problems in arithmetic geometry is to understand the set V (k) of rational points on an algebraic variety V defined over a number field k. When V = E is an elliptic
Quadratic twists of pairs of elliptic curves
Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^\times$ with pairwise distinct image in $ K^\times/{K^\times}^2
Isogenies of supersingular elliptic curves over finite fields and operations in elliptic cohomology
We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a
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References

SHOWING 1-10 OF 215 REFERENCES
Elliptic Curves: Notes from Postgraduate Lectures Given in Lausanne 1971/72
Complex Elliptic Curves.- Elliptic Curves in Characteristic zero.- Division Points.- Complements.- Complex Elliptic Curves.- Elliptic Curves in Characteristic zero.- Division Points.- Complements.
Elliptic Curves: Diophantine Analysis
I. General Algebraic Theory.- I. Elliptic Functions.- II. The Division Equation.- III. p-Adic Addition.- IV. Heights.- V. Kummer Theory.- V1. Integral Points.- II. Approximation of Logarithms.- VII.
Introduction to the arithmetic theory of automorphic functions
* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary
Arithmetic moduli of elliptic curves
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"
Elliptic curve cryptosystems
TLDR
The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.
A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
The two fundamental finiteness theorems in the arithmetic theory of elliptic curves are the Mordell-Weil theorem, which says that the group of rational points is finitely generated, and Siegel's
On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields
  • G. Shimura
  • Mathematics
    Nagoya Mathematical Journal
  • 1971
1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z).
Abelian curves of 2-power conductor
  • A. Ogg
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1966
Let Q be the field of rational numbers, and let A be an Abelian curve (an Abelian variety of dimension one) defined over Q. Following Weil, the conductor of A is where p runs over all primes, and the
Large Integral Points on Elliptic Curves
To ml, friend Dan Shanks Abstract. We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, ,') of certain
A Course in Arithmetic
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on
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