The Arithmetic of Elliptic CurvesSpringer Science & Business Media, 20 avr. 2009 - 513 pages 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 approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics. |
Table des matières
1 | |
17 | |
III The Geometry of Elliptic Curves | 41 |
IV The Formal Group of an Elliptic Curve | 115 |
V Elliptic Curves over Finite Fields | 137 |
VI Elliptic Curves over C | 157 |
VII Elliptic Curves over Local Fields | 184 |
VIII Elliptic Curves over Global Fields | 207 |
XI Algorithmic Aspects of Elliptic Curves | 362 |
A Elliptic Curves in Characteristics 2 and 3 | 409 |
B Group Cohomology H0 and H1 | 415 |
An Overview | 425 |
Notes on Exercises | 458 |
List of Notation | 467 |
472 | |
489 | |
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Expressions et termes fréquents
action addition algebraic algorithm apply associated assume assumption bound called Chapter char(K choose coefficients complete compute conjecture consider constant contains coordinates corresponding defined Definition denote depending described differential discriminant divisor E(Fq element elliptic curve End(E equal equivalence exact example Exercise exists extension fact factor field Finally finite fixed formal group formula function Further given gives GR/k height Hence homogeneous homomorphism ideal implies integer isogeny isomorphism Let E/K minimal modular modulo morphism multiplicative natural Note obtain pairing particular polynomial prime problem PROOF properties Proposition Prove quadratic rational reduction Remark respectively result ring root satisfying says sequence smooth solutions space steps subgroup Suppose tells theorem theory variety Weierstrass equation write yields