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Zur Arithmetik der elliptischen Kurven. (Apropos of the arithmetic of the elliptic curves). (German) Zbl 0619.14021

Ber. Math.-Stat. Sekt. Forschungsges. Joanneum 271, 107 p. (1986).
The author provides a nice summary of the arithmetic theory of elliptic curves. Starting with the basic formulae associated to Weierstraß equations, he describes the main theorems concerning elliptic curves over finite, local, and global fields. Further topics covered include torsion subgroup, height functions, integral points, Galois cohomology, L-series, the conjecture of Birch and Swinnerton-Dyer, modular curves, and Heegner points. The author concludes with a detailed discussion of the theorems of Goldfeld and Gross-Zagier which together provide a solution to Gauss’ class number problem. The author makes no attempt to prove the main theorems, but gives extensive references. In this way, he is able to provide in 100 pages an easily read summary of a fascinating theory.
In short, this monograph is a good introduction to an interesting subject, similar in scope to the survey article of J. T. Tate [Invent. Math. 23, 179-206 (1974; Zbl 0296.14018)], but containing many more details and highlighting the many exciting discoveries of the last decade.
Reviewer: J.Silverman

MSC:

14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
14H25 Arithmetic ground fields for curves
14G15 Finite ground fields in algebraic geometry
14G20 Local ground fields in algebraic geometry
14G25 Global ground fields in algebraic geometry

Citations:

Zbl 0296.14018