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). 
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3,054 Citations

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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…
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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

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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
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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"…
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TLDR
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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).…
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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
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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