# The arithmetic of elliptic curves

@inproceedings{Silverman1986TheAO, title={The arithmetic of elliptic curves}, author={Joseph H. Silverman}, booktitle={Graduate texts in mathematics}, year={1986} }

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).

## 3,054 Citations

THE GEOMETRY OF ELLIPTIC CURVES OVER FINITE FIELDS

- Mathematics
- 2016

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

- Mathematics, Computer Science
- 2011

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…

Some remarks on arithmetical properties of recursive sequences on elliptic curves over a finite field

- Mathematics, Computer Science
- 2007

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.

A Combinatorial Exploration of Elliptic Curves

- Mathematics, Computer Science
- 2015

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

- MathematicsDes. Codes Cryptogr.
- 2010

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

- Mathematics
- 2015

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

- Mathematics

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

- Mathematics
- 2010

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

- Mathematics
- 2006

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

- Mathematics
- 2007

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…

## References

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- Mathematics
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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.

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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…

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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

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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

- Mathematics
- 1987

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

- Mathematics
- 1973

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…